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no.06--28
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http://www.archive.org/details/learningdisagreeOOacem
DEWEY
HB31
.M415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
LEARNING AND DISAGREEMENT
UNCERTAIN WORLD
IN
AN
Daron Acemoglu
Victor Chernozhukov
Muhammad
Yildiz
Working Paper 06-28
October 20, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research Network
Paper Collection
http://ssrn.com/abstract=9391 69
at
'""iEii?.Li&-^
J=!§RmiEs
Learning and Disagreement in an Uncertain World*
Daron Acemoglu, Victor Chernozhukov, and Muhamet
Yildiz^
October, 2006.
Abstract
Most economic analyses presume that there are limited differences in the prior beliefs of
most often justified by the argument that sufficient common experiences and observations will eliminate disagreements. We investigate this claim using a
simple model of Bayesian learning. Two individuals with different priors observe the same
infinite sequence of signals about some underlying parameter. Existing results in the literature
individuals, an assumption
establish that
when
individuals are certain about the interpretation of signals, under very mild
conditions there will be asymptotic agreement
—
their assessments will eventually agree. In con-
we look
at an environment in which individuals are uncertain about the interpretation of
meaning that they have non-degenerate probability distributions over the conditional
distribution of signals given the underlying parameter. When priors on the parameter and the
trast,
signals,
conditional distribution of signals have
full
we prove the following results: (1) Indisame infinite sequence of signals. (2) Before
support,
viduals will never agree, even after observing the
observing the signals, they believe with probability
1
that their posteriors about the underlying
(3) Observing the same sequence of signals may lead to a divergence of opinion rather than the typically-presumed convergence. We then characterize the
parameter
will fail to converge.
conditions for asymptotic agreement under "approximate certainty"
limit
—
i.e.,
where uncertainty about the interpretation of the signals disappears.
as
we
look at the
When
the family
of probability distributions of signals given the parameter has "rapidly-varying tails" (such
as the
normal or the exponential distributions), approximate certainty restores asymptotic
agreement. However, when the family of probability distributions has "regularly-varying
(such as the Pareto, the log-normal, and the t-distributions)
,
tails"
asymptotic agreement does not
obtain even in the limit as the amount of uncertainty disappears.
Lack of
common
circumstances.
We
economic behavior
priors has important implications for
illustrate
economic behavior
in a range of
the type of learning outlined in this paper interacts with
in various different situations, including
nation, asset trading
games
of
common
interest, coordi-
and bargaining.
Keywords: asymptotic
JEL
how
disagreement, Bayesian learning, merging of opinions.
Classification: Oil, C72, D83.
*We thank Eduardo
Faingold, Greg Fisher,
Drew Fudenberg, Alessandro
Lizzeri,
Giuseppe Moscarini, Mar-
ciano Sinscalchi, Robert Wilson and seminar participants at the University of British Columbia, University of
Urbana-Champaign and Yale for useful comments and suggestions.
Department of Economics, Massachusetts Institute of Technology.
Illinois at
*
Introduction
1
The common
prior assumption
is
one of the cornerstones of modern economic analysis. Most
in a
models postulate that the players
precisely, that they
for
example, they
known
have a
all
common
agree that
game have the "same model
prior about the
some
6.
The
may
own
common
prior
view
is
among
d,
p.
A
some
this process will lead to
A
6.
strong version of this
48) as the statement that a Bayesian individual,
does not assign zero probability to "the truth,"
are informative about the truth.
drawn from a
assumption comes from
and
individuals about the distribution of the vector
expressed in Savage (1954,
is
experiences and the communication of others, will
have access to a history of events informative about the vector
"agreement"
more
distributions
also have additional information about
typical justification for the
learning; individuals, through their
game form and payoff
payoff-relevant parameter vector 6
distribution G, even though each
components of
of the world," or
will learn
who
eventually as long as the signals
it
more sophisticated version
of this conclusion also follows
from Blackwell and Dubins' (1962) theorem about the "merging of opinions".^
Despite these powerful intuitions and theorems, disagreement
exception in practice. Just to mention a few instances, there
ment even among economists working on a certain
topic.
is
the rule rather than the
typically considerable disagree-
is
For example, economists routinely
disagree about the role of monetary policy, the impact of subsidies on investment or the mag-
nitude of the returns to schooling.
Similarly, there are deep divides
within populations with shared experiences, and
finally,
there
about religious
beliefs
was recently considerable
dis-
agreement among experts with access to the same data about whether Iraq had weapons of
mass destruction. In none of these
cases,
can the disagreements be traced to individuals having
access to different histories of observations. Rather
particular,
it
it is
their interpretations that differ.
seems that an estimate showing that subsidies increase investment
is
In
interpreted
very differently by two economists starting with different priors; for example, an economist
believing that subsidies have no effect on investment appears
more
likely to
judge the data or
the methods leading to this estimate to be unreliable and thus to attach less importance to this
evidence. Similarly, those
who
believed in the existence of weapons of mass destruction in Iraq
Blackwell and Dubins' (1962) theorem shows that
two probability measures are absolutely continuous with
same events), then as the number of
observations goes to infinity, their predictions about future frequencies will agree. This is also related to Doob's
(1948) consistency theorem for Bayesian posteriors, which we discuss and use below.
if
respect to each other (meaning that they assign positive probability to the
presumably interpreted the evidence from inspectors and journalists indicating the opposite
as
biased rather than informative.
In this paper,
we show that
type of behavior will be the outcome of learning by
this
Bayesian individuals with different priors when they are uncertain about the informativeness
of signals. In particular,
we consider the
following simple environment: one or two individuals
with given priors observe a sequence of signals, {st}"^Q, and form their posteriors about some
underlying state variable (or parameter)
is
may be
that these individuals
9.
The only non-standard
uncertain about the distribution of signals conditional on
In the simplest case where the state
9 e {A,B}, and
{a, &}, this implies that Pr(st
and the
the underlying state.
st
G
=
9
individuals' subjective probability distribution
This distribution, which can
differ
and to
among
uncertainty about the informativeness of signals.
are non- degenerate, individuals will have
some
its
9)
\
We
but individuals also have a prior over p0, say given by Fg.
density.
feature of the environment
—
pg
signal are binary, e.g.,
is
not a
known number,
refer to this distribution
Fg as
density fg as subjective (probability)
individuals,
When
is
a natural measure of their
subjective probability distributions
latitude in interpreting the sequence of signals
they observe.
We
identify conditions
under which Bayesian updating leads to asymptotic learning
(indi-
viduals learning, or believing that they will be learning, the true value of 9 with probability
after observing infinitely
many
assessments of the value of
9).
signals)
We
asymptotic learning and agreement.
and asymptotic agreement (convergence between
first
are guaranteed.
probability
1
p'
>
Second,
First,
we show that
1/2 (with possibly p^
we
learning under certainty leads to as-
^ p^),
>
i is
sure that pg
= p'
for
then asymptotic learning and agreement
establish the stronger results that
to the event that pg
their
provide some generalizations of existing results on
ymptotic learning and agreement. In particular, when each individual
some known number
1
when both
individuals attach
1/2 for 9 G {A, B}, then there will again be asymptotic
learning and agreement.
These positive
be
less
1.
results
do not hold, however, when there
than 1/2. In particular, when Fg has a
There
will not
full
is
support
positive probability that pg might
for
each
9,
we show
that:
be asymptotic learning. Instead each individual's posterior of 9 continues
to be a function of his prior.
2.
There
will
not be asymptotic agreement; two individuals with different priors observing
of signals will reach different posterior beliefs even after observing
same sequence
the
infinitely
many
signals.
Moreover, individuals attach ex ante probability
1
that they will
disagree after observing the sequence of signals.
3.
Two
individuals
may
disagree
they did so previously. In
the
full
more
fact, for
it
common
sequence of signals than
any model of learning under uncertainty that
satisfies
support assumption, there exists an open set of pairs of priors such that the
disagreement between the two individuals
While
a
after observing
may appear
the event that pe
<
will necessarily
grow starting from these
priors.
plausible that the individuals should not attach zero probability to
1/2,
it
is
also reasonable to expect that the probability of such events
should be relatively small. This raises the question of whether the results regarding the lack
of asymptotic learning
of uncertainty.
Put
and agreement
results
We
and agreement under uncertainty survive when there
differently,
we would
like to
is
a small
understand whether the asymptotic learning
under certainty are robust to a small amount of uncertainty.
investigate this issue by studying learning under "approximate certainty,"
sidering a family of subjective density functions {fm} that
around a single point
amount
—thus converging to
i.e.,
by con-
become more and more concentrated
full certainty.
It is
straightforward to see that as
each individual becomes more and more certain about the interpretation of the signals, as-
ymptotic learning obtains. Interestingly, however, the conditions
much more demanding than
those for cisymptotic learning.
for
asymptotic agreement are
Consequently, even though each
individual expects to learn the payoff-relevant parameters, asymptotic agreement
obtain. This implies that asymptotic agreement under certainty
point of a general model of learning under uncertainty.
is
the case depends on the
tail
this family has regularly-varying tails (such as the
tions),
even under approximate certainty there
to
a discontinuous limit
show that whether or not
this
will
Pareto or the log-normal distribu-
be asymptotic disagreement.
When {fm}
(such as the normal distribution), there will be asymptotic agreement
has rapidly- varying
tails
under approximate
certainty.
approximate certainty
will learn the payoff- relevant
will fail to learn.
fail
properties of the family of subjective density functions {fm}-
When
Intuitively,
We
may be
may
Whether
is
sufficient to
parameter, but they
or not they believe this
make each
may
still
individual believe that they
believe that the other individual
depends on how an individual reacts when
a frequency of signals different from the one he expects with "almost certainty" occurs.
If
this event prevents the individual
under approximate certainty.
trusts his
own model
with his model.
This
of the world
When
from learning, then there
will
be asymptotic disagreement
because under approximate certainty, each individual
is
and thus expects the limiting frequencies to be consistent
the other individual's model of the world differs, he expects the other
Then whether
individual to be surprised by the limiting frequencies of the signals.
asymptotic agreement
will
obtain depends on whether this surprise
other individual from learning, which in turn depends on the
tail
is
or not
sufficient to prevent the
properties of the family of
subjective density functions {/m}-
Lack of asymptotic agreement has important implications
tions.
We
illustrate
some
individuals observe the
we
of these
for a
by considering a number of simple environments where two
same sequence
of signals before or while playing a game. In particular,
games
discuss the implications of learning in uncertain environments for
games
of
common
interest, bargaining,
games
of
than they would
in
environments with
common
priors
will play these
establish that contrary to standard results, individuals
more information about
observing the same sequence of signals
may
games
— and also differently than
ments without common priors but where learning takes place under
interests before receiving
of coordination,
payoffs.
certainty.
in environ-
For example,
may
wish to play games of
We
show how the
also
common
possibility of
lead individuals to trade only after they observe
Stokey, 1982) and existing results on asset trading without
common
provide a simple example illustrating a potential reason
why
individuals
(e.g.,
priors,
assume learning under certainty (Harrison and Kreps, 1978, and Morris, 1996).
about informativeness of signals
show
differently
the public information. This result contrasts with both standard no-trade theorems
grom and
We
communication and asset trading.
how, when they are learning under uncertainty, individuals
we
range of economic situa-
Mil-
which
Finally,
may be
we
uncertain
—the strategic behavior of other agents trying to manipulate
their beliefs.
Our
results cast
learning. In
many
doubt on the idea that the common prior assumption may be
environments, even
when
there
is little
justified
by
uncertainty so that each individual
believes that he will learn the true state, learning does not necessarily imply agreement about
the relevant parameters.
Consequently, the strategic outcome
from that of the common-prior environment.^ Whether
a
this
may be
assumption
significantly different
is
warranted therefore
For previous arguments on whether game-theoretic models should be formulated with all individuals having
prior, see, for example, Aumann (1986, 1998) and Gul (1998). Gul (1998), for example, questions
common
depends on the
specific setting
and what type of information individuals are trying to glean
from the data.
Relating our results to the famous Blackwell-Dubins (1962) theorem
essence.
As
briefly
mentioned
on zero-probability events
in
(i.e.,
other), asymptotically, they will
Our
results
do not contradict
Footnote
1,
may
theorem shows that when two agents agree
this
their priors are absolutely continuous
make the same
this
help clarify their
with respect to each
predictions about future frequencies of signals.
theorem, since we impose absolute continuity throughout.
Instead, our results rely on the fact that agreeing about future frequencies
not the same
is
Put
as agreeing about the underlying state (or the underlying payoff relevant parameters).'^
differently,
under uncertainty, there
is
an "identification problem" making
it
impossible for
individuals to infer the underlying state from limiting frequencies and this leads to different
interpretations of the
nomic
situations,
same
what
is
signal sequence
important
is
by individuals with different
priors.
most
In
eco-
not future frequencies of signals but some payoff-relevant
parameter. For example, what was essential for the debate on the weapons of mass destruction
was not the frequency
is
of
news about such weapons but whether or not they
relevant for economists trying to evaluate a policy
effect of similar policies
(and
if)
is
may be
What
not the frequency of estimates on the
from other researchers, but the impact of
implemented. Similarly, what
existed.
this specific policy
relevant in trading assets
is
when
not the frequency
of information about the dividend process, but the actual dividend that the asset will pay.
Thus, many situations
will
which individuals need to learn about a parameter or state that
in
determine their ultimate payoff as a function of their action
falls
within the realm of the
analysis here.
In this respect, our work differs from papers, such as
Freedman
(1963, 1965)
and Miller
and Sanchirico (1999), that question the applicability of the absolute continuity assumption
in the
Blackwell-Dubins theorem in statistical and economic settings (see also Diaconis and
Freedman, 1986, Stinchcombe, 2005).
for
Similarly, a
number
of important theorems in statistics,
example. Berk (1966), show that under certain conditions, limiting posteriors
their support
on the
limiting distribution).
whether the
common
set of all identifiable values
Our
results are different
prior assumption maizes sense
'in this respect, our paper
is
also related to
agree about long-run frequencies, but their beliefs
fail
may
fail
there
is
no ex ante
(1994, 1996),
to
who
merge because
have
to converge to a
from those of Berk both because
when
Kurz
(though they
will
in
our model
stage.
considers a situation in which agents
of the non-stationarity of the world.
individuals always place positive probability on the truth and also because
characterization of the conditions for lack of asymptotic learning
Our paper
also closely related to recent independent
is
"common knowledge" when
signals.
They show how
the signal space
is infinite.
how
will
Ely,
Mailath and
be "common learning" by
individual learning
may
not lead to
Cripps, Ely, Mailath and Samuelson's
analysis focuses on the case in which the agents start with
certainty (though they note
tight
and agreement.
work by Cripps,
Samuelson (2006), who study the conditions under which there
two agents observing correlated
we provide a
common
priors
and learn under
can be extended to the case of non-common
their results
Consequently, our emphasis on learning under uncertainty and the results on learning
priors).
under conditions of approximate certainty are not shared by this paper. Nevertheless, there
is
a close connection between our result that under approximate certainty each agent expects
that he will learn the payoff-relevant parameters but that he will disagree with the other agent
and Cripps,
Ely,
common
Mailath and Samuelson's finding of lack of
learning with infinite-
dimensional signal spaces.
The
rest of the
paper
is
organized as follows. Section 2 provides
all
our main results in the
context of a two-state two-signal setup. Section 3 provides generalizations of these results to
an environment with
of our results,
K states and L> K signals.
and Section
Section 4 considers a variety of applications
5 concludes.
The Two-State Model
2
Environment
2.1
We
model with binary
start with a two-state
signals.
This model
is
sufficient to establish all
our main results in the simplest possible setting. These results are generalized to arbitrary
number
of states
and
signal values in Section
There are two individuals, denoted by
{st}"_Q where
ability
i.e.,
tt'
6
£
St
(0, \)
{a, b].
to 9
=
The underlying
A.
The
i
3.
=
state
1
is
and
^
i
—
2,
who
observe a sequence of signals
e {A, B], and agent
individuals believe that, given
9,
i
assigns ex ante prob-
the signals are exchangeable,
they are independently and identically distributed with an unknown distribution.^ That
"See, for example, Billingsley (1995). If there were only one state, then our
model would be
identical to
De
example, Savage, 1954). In the context of this model, De Finetti's theorem
provides a Bayesian foundation for classical probability theory by showing that exchangeability (i.e., invariance
under permutations of the order of signals) is equivalent to having an independent identical unknown distrib-
Finetti's canonical
model
(see, for
ution and implies that posteriors converge to long-run frequencies.
De
Finetti's decomposition of probability
the probability of
is,
of
=
Si
6
given
=5
6*
=
sj
is
= A
a given 9
is B.n
unknown number p^;
an unknown number pB
—
as
b
PA
1
1
-PA
-PB
PB
Our main departure from the standard models
uncertain about pA and
to individual
is
i
—
We
pB
is
we allow the
that
individuals to be
denote the cumulative distribution function of po according
his subjective probability distribution
i.e.,
in the following table:
B
A
a
shown
likewise, the probability
—by Fg.
In the standard models, F^
degenerate and puts probability 1 at some pg. In contrast, for most of the analysis,
we
will
impose the following assumption:
Assumption
For each
1
i
and
9,
Fg has a continuous, non-zero and finite density fg over
[0,1].
The assumption
implies that Fg has full support over
assumption allows Fg
this
whether or not
this
is
so
the distribution o( pg).
(p)
(i.e.,
and Fg
(p) to differ, for
[0, 1].
many
worth noting that while
It is
of our results
Remark
2,
not important
common
whether or not the two individuals have a
Moreover, as discussed in
it is
Assumption
1
is
prior about
stronger than
necessary for our results, but simplifies the exposition.
In addition, throughout
We
we assume that
consider infinite sequences s
The
sequences.
=
7r\
tt^,
Fg and Fg are known to both individuals.^
{sf}^^ of signals and write
posterior belief of individual
i
S
for the set of all
about 9 after observing the
first
n
such
signals {st]^^^
is
where Pr^
{9
=A
\
{sf}JLj) denotes the posterior probability that 9
signals {sf}"^i under prior
tt*
— A
and subjective probability distribution Fg
given a sequence of
(see footnote 7 for a
formal definition).
Throughout, without
loss of generality,
we suppose that
in reality 9
=
A.
The two questions
of interest for us are:
extended by Jackson, Kalai and Smorodinsky (1999) to cover cases without exchangeability.
player 1 knows the prior and probability assessment of player 2 regarding the distribution of signals given the state is used in the "asymptotic agreement" results and in applications. Since our
purpose is to understand whether learning justifies tlie common prior assumption, we assume that agents do
not change their views because the beliefs of others differ from theirs.
distributions
°
is
The assumption that
1.
Asymptotic
2.
Asymptotic agreement: whether
lecirning: whether Pr'
(Hm„^oo
=
(Ki (s)
Pr' (Hm„_,oo
l\9
\'Pn i^)
~
=
A)
'Pn{^)\
=
—
1 for
O)
z
=
=
1, 2.
1 for z
=
1, 2.
Notice that both asymptotic learning and agreement are defined in terms of the ex ante
probabihty assessments of the two individuals. Therefore, asymptotic learning implies that an
individual, believes that he or she will ultimately learn the truth, while asymptotic
agreement
implies that both individuals believe that their assessments will eventually converge.^
Asymptotic Lecirning and Disagreement
2.2
The
following theorem gives the well-known result, which applies
A
hold.
version of this result
is
when Assumption
1
does not
stated in Savage (1954) and also follows from Blackwell and
Dubins' (1962) more general theorem applied to this case.
uses different arguments than those presented below and
Since the proof of this theorem
tangential to our focus here,
is
it is
relegated to the Appendix.
Theorem
Fg
(p')
=
1
1
Assume
and Fg
that for
(p)
=
some
p^,fP'
for each p
=
=
A)
<
=
p^.
£
Pr^(lim„_oo<^Us)
2.
Fv' (lim„^oo 1^^ (s) -<t>lis)\=0)==l.
Theorem
1 is
will learn the
the
i
—
1
on p%
i.e.,
1,2,
l.
a slightly generalized version of the standard theorem where the individual
truth with experience (almost surely as
same sequence
will necessarily agree.
The
that p^
=
follow.
The theorem
The
the probability oi
intuition of this
st
theorem
states that even
=
if
is
^
oo)
and two individuals observing
is
differ
from p^ (while Savage, 1954, assumes
useful for understanding the results that will
the two individuals have different expectations about
a conditional on 9
these beliefs with certainty
n
generalization arises from the fact that learning
and agreement take place even though p^ may
p~).
each Fg puts probability
Then, for each
1.
l|0
(1/2,1],
sufficient for
=
A, the fact that p*
>
1/2 and that they hold
asymptotic learning and agreement. For example,
We formulate asymptotic learning in terms of each individual's initial probability measure so as not to take
a position on what the "objective" for "true" probability measure is.
In terms of asymptotic agreement, we will see that Pr' (lim„_oo {(p^ (*) "~ "^n (*)| — O) = 1 also implies
=0
limn—oo |0j, (s) — 4>'^{s)\
for almost all sample paths, thus individual
totic agreement coincide with asymptotic agreement (and vice versa).
beliefs that there will
be asymp-
consider an individual
=
6
is
A.
1
p\
in
learning applies
which case he
will
—
he will conclude that 6
that this individual
is
=
conclude that
the true state with probability
9
is
=
Given that p >
and
sufficient to observe that this
Next to see why asymptotic agreement obtains, suppose
1.
confronted with a frequency p
>
p^ of
when
a signals. Since he believes with
the state
9
is
=A
and
1
—
p''
when
B, he will interpret the frequency p as resulting from samphng variation.
p^, this
sampling variation
1
much much more
is
frequency of a signals p'
when
even though p^
the state
^ fP
is
^
=A
—
A.
the state
is
9
^A
Asymptotic agreement
believes that individual j will observe a
and expects that he
and
(as long as p'
i
when
likely
to the event that 9
then follows from the observation that individual
=A
the underlying state
B. Therefore, this individual believes that he will learn
therefore, he will attach probability
that 9
is
when
1/2
A, or he will observe a limiting frequency of
certainty that the frequency of signals should be p*
the state
it
>
sure that he will be confronted either with a limiting frequency of a signals equal
is
— p', and
expects a frequency of a signals p*
why asymptotic
First, to see
individual
to
who
fp are
will
conclude from this
both greater than 1/2 as assumed
in
the theorem).
We
next generalize Theorem
1
about the signal distribution but
to the case
where the individuals are not necessarily certain
their subjective distributions
do not
satisfy the full
support
feature of Assumption 1
Theorem
i
=
2
Assume
that each Fq has a density fg
and
satisfies
Fg (1/2)
—
0.
Then, for each
l,2,
1.
Pr'(lim„^oo<^Us)
2.
Pr' (lim„^oo
This theorem
Fq (1/2)
The
=
\4>l is)
will
=
l|e
-<pl{s)\^Q)^l.
be proved together with Theorem
implies that pg
intuition for this result
probability
1
= A)-l.
>
is
3.
It is
evident that the assumption
1/2, contradicting the full support feature in
similar to that of
to the event that pg
the limiting distribution will be
>
Sj
Theorem
1:
when both
Assumption
1.
individuals attach
1/2, they will believe that the majority of the signals in
=
a
when
6
=
A.
Thus, each believes that both he and
the other individual will learn the underlying state with probability
1
(even though they
both be uncertain about the exact distribution of signals conditional on the underlying
may
state).
This theorem shows that results on asymptotic learning and agreement are substantially more
9
Nevertheless, the result relies on the feature that
general than Savage's original theorem.
=
F^ (1/2)
each
for
i
=
1,2
and each
This implies that both individuals attach zero
6.
probabihty to a range of possible models of the world
be
less
than 1/2.
each individual
may
It
may
instead be
—
more reasonable
i.e.,
to
they are certain that pe cannot
presume
under uncertainty,
that,
attach positive (though perhaps small) probabihty to
encapsulated by Assumption
1.
We next
general circumstances where Fg has
full
impose
this
all
values of pg as
assumption and show that under the more
support, there will be neither asymptotic learning nor
asymptotic agreement.
Theorem
3 Suppose Assumption
1.
Pr' (lim„^oo
2.
Pr' (lim„-,oo
4^n
\<l>li
1 holds
{s)^l\e^A)=l
(s)
-
(t>n
{s)\¥'^)
for
for
=
1
i
i
=
=
1,2.
Then,
1,2;
whenever
tt^
^
tt^
and F^
= F^
e
for each
{A,B}.
This theorem contrasts with Theorems
will fail to learn the true state
if
1
and
with probability
2
1.
and implies that the individual
The second
in question
part of the theorem states that
the individuals' prior beliefs about the state differ (but they interpret the signals in the
same way), then
probability
that there
1
is
their posteriors will eventually disagree,
and moreover, they
to the event that their beliefs will eventually diverge.
Put
both attach
differently, this implies
"agreement to eventually disagree" between the two individuals, in the sense that
they both believe ex ante that after observing the signals they will
will
will
to agree. This feature
fail
play an important role in the applications in Section 4 below.
Remark
1
The assumption
that Fg
—
Fg
in this
theorem
is
adopted
for simplicity.
Even
in
the absence of this condition, there will typically be no asymptotic agreement. Theorem 6 in
the next section generalizes this theorem to a situation with multiple states and multiple signals
and
also dispenses with the
assumption that Fg
—
Fg.
It
establishes that the set of priors
subjective probability distributions that leads to asymptotic agreement
Remark
2 Assumption
only for simplicity.
It
1 is
considerably stronger than necessary for
is
of "measure zero".
Theorem
can be verified that for lack of asymptotic learning
it is
3 and
is
adopted
sufficient (but not
necessary) that the measures generated by the distribution functions F\'{p) and
10
and
Fg
(1
—
p) be
absolutely continuous with respect to each other. Similarly, for lack of asymptotic agreement,
it is
not necessary) that the measures generated by
sufficient (but
and Fq
(1
-
that
pb
is
pA
is
(p),
F^
(1
For example,
p) be absolutely continuous with respect each other.
viduals beheve that
F\
—
if
p),
F^
both
either 0.3 or 0.7 (with the latter receiving greater probability)
(p)
indi-
and
also either 0.3 or 0.7 (with the former receiving greater probability), then there will
be neither asymptotic learning nor asymptotic agreement. Throughout we use Assumption
both because
it
and because
simplifies the notation
a natural assumption
it is
1
when we turn
to the analysis of asymptotic agreement under approximate certainty below.
Towards proving the above theorems, we now introduce some notation, which
throughout the paper. Recall that the sequence of
signals, s,
is
will
be used
generated by an exchangeable
process, so that the order of the signals does not matter for the posterior. Let
= H^{t<
rn{s)
be the number of times
r-n (s)
St
=
/n converges to some p
a out of
(s)
e
first
n
n\st
=
a)
By
signals.^
the strong law of large numbers,
almost surely according to both individuals. Defining
[0, 1]
the set
S=
{s e
this observation implies that Pr' (s
sample paths
G
S
lim„-,oo Tn (s)
:
=
S")
1
for
i
=
/n
1, 2.
exists}
We
Now, a straightforward application
or with probability
1
where Pr'
the probability of observing the signal
is
.
^
(s)
=
lim„^oo
i'li
(5) for all
Given the definition of r^
=
sample paths
-
(b^'^-)
=
{(9,s')
\s't
St
our results for
all
=
st for
""'
(1
each
[
-
P""*'* (^
Tr')
(1
"
=
a exactly r„ times out of
-
<
p)'-"^^'
n}, where s
11
B} x S
=
p-^-'^'/b
\s
{st}^i and
s'
(p)
=
n
useful formula for
P)""""*"' Ja (p) dp, and
I'
t
Bayes rule gives
s in S.
the probabihty distribution Pr' on {A,
Pr' (e^'"''^'^
Pr'
at each event e"'"'"
{s\,
of the
The next lemma provides a very
signals with respect to the distribution Fq.
(j!)'^
will often state
which equivalently implies that these statements are true almost surely
s in S,
(r„|6')
(1)
,
dp
{sjjj^j.
Lemma
1
Suppose Assumption
=
<Pl [p (s))
where p
(s)
=
lim„_»oo ^n (s) /n,
Then for
holds.
1
4
lim
and Wp G
all s
E S,
=
[s)
}
(3)
[0, 1],
Proof. Write
W{rn\e^B)
Pr' irn\e = A)
J^p--{l-pr-'-fB{l-p)dp
^
JoP'H^-Pr-'-fA{p)dp
/q p''"(l-p)"-'""dp
~
/n'p'-"(l-p)"-'-"/A(p)dp
E^lfBJl
Here, the
first
same term.
Jb
(1
—
{sf}"^Q.
equaUty
The
obtained by dividing the numerator and the denominator by the
is
resulting expression on the numerator
p) given r„
Denoting
- p)\rn]
under the
this
by
the conditional expectation of Ja
the conditional expectation of
(Lebesgue) prior on p and the Bernoulli distribution on
flat
— p)|r„], and
E'^[/j3(l
is
(p),
by
the denominator, which
we obtain the
E'**[/4(p)|r„],
is
similarly defined as
last equality.
consistency theorem for Bayesian posterior expectation of the parameter, as rn -+
that E^[/b(1 -p)|r„]
^
/s(l
-
and E^[/A(p)|r„]
p)
and Ramamoorthi, 2003, Theorem
1.3.2).
^
/^(p)
(see, e.g.,
By Doob's
p,
we have
Doob, 1949, Ghosh
This establishes
PTHrn\e
=
B)
,
as defined in (4). Equation (3) then follows from (2).
In equation (4), ff (p)
the true state
is
B
versus
this likelihood ratio
An immediate
proofs of
the asymptotic likelihood ratio of observing frequency p of a
when
and Bayes
it is
=
4>lo
Theorems
2
A.
rule to
implication of
4>l iP is))
The
is
(P is))
Lemma
compute
Lemma
if
1
1 is
and only
states that, asymptotically, individual
his posterior beliefs
about
when
i
uses
6.
that given any s & S,
if
^^i?^ [p (s)) = ^—^R" {p (s))
and 3 now follow from
12
Lemma
1
and equation
(5).
(5)
Proof of Theorem
ment
Lemma
in
when p (s) <
1
I
for
=
=
z
1 still applies,
Given 9
1/2.
cording to both
Under the assumption that Fg
2.
i
== 1
and
—
2.
and we have
A, then r„
Hence, Pr'
Similarly, Pi'
1,2.
=
=
{p (s))
{(j)]^
=
=
0\e
theorem, the argu-
when p (s) > 1/2 and K'
=
l\e
B)
in the
=
A)
=
(s)
>
=
{p (s))
oo
1/2 almost surely ac-
=
Pr' (<^^ (p (s))
=
Pr' (<^^ (p(s))
O|0
1\0
=
A)
= B) =
=
for
1
1,2, establishing the second part.
Proof of Theorem
Hence, by
Lemma
1,
Since fg (1
3.
(f>^^
(p(s))
¥"
follows from equation (5), since n^
and thus Pr'
rf>l is),
Intuitively,
is
{p (s))
=
/n converges to some p
(s)
{p {s))
{(f)]^
i?'
(1/2)
{\cf>l [s)
- ^^
when Assumption
-
>
p{s))
and /a (p(s))
is
finite,
for each s, establishing the first part.
1
^
tv'
(s)|
1 (in
=
and Fq
^
O)
=
1
Fq implies that
for
i
=
for
each
>
R' (p(s))
The second
s
E S,
part
(f)^ (s)
his views
particular, the full support feature) holds,
about pe (the informativeness of the signals) as well as
underlying state.
For example, even when signal a
B, a very high frequency of a
because he
may
is
more
an individual
a.
about the state and
(3),
i?' (p)
us
tells
how
way
informativeness of the signals in the same
always be reflected
In contrast,
if
were sure that PA
he
initially
in state
is
beheved, and they
be determined by
A,
may
his prior beliefs
(i.e.,
R^
When
—
two individuals interpret the
R^), the differences in their priors
an individual were sure about the informativeness of the signals
— PB =
to a value different from
^
than
in their posteriors.
P^ for
some
p*
question the informativeness of the signals
p{s)
A
the individual updates his beliefs about the
informativeness of the signals as he observes the signals.
will
views about the
defined in (4) never takes the value zero or infinity.
his posterior beliefs will
by R^, which
also
likely in state
will
Therefore, the individual never becomes certain about the state,
captured by the fact that
Consequently, as shown in
his
not necessarily convince him that the true state
infer that the signals are not as reliable as
instead be biased towards
which
will
is
^
1, 2.
never sure about the exact interpretation of the sequence of signals he observes and
update
0.
p'or
1
—
p'.
>
1/2) as in
Theorem
1,
(i.e.,
if i
then he would never
—even when the limiting frequency
of a converges
Consequently, in this case, for each sample path with
1/2 both individuals would learn the true state and their posterior beliefs would agree
asymptotically.
As noted above, an important implication
of
Theorem
3
is
that there will tj^aically be
"agreement to eventually disagree" between the individuals. In other words, given their
13
priors,
both individuals
the same infinite sequence of signals they will
will agree that after seeing
disagree (with probability
This implication
1).
is
interesting in part because the
common
still
prior
assumption, typically justified by learning, leads to the celebrated "no agreement to disagree"
result
(Aumann,
1976, 1998), which states that
if
the individuals' posterior beliefs are
common
knowledge, then they must be equal.^ In contrast, in the limit of the learning process here,
individuals' beliefs are
common knowledge
different with probability
as in
Assumption
beliefs
1.
This
is
(as there
is
because in the presence of uncertainty and
both individuals understand that
1,
no private information), but they are
support
full
on
their priors will have an effect
even asymptotically; thus they expect to disagree.
Many
of the applications
we
their
discuss
in Section 4 exploit this feature.
Divergence of Opinions
2.3
Theorem
3 established that the differences in priors are reflected in the posteriors even in
the limit as
n —+
oo.
two individuals. The
first
does not, however, quantify the possible disagreement between the
rest of this section investigates different aspects of this question.
show that two individuals that observe the same sequence of
posteriors, so that
Theorem
satisfy
p G
It
common
4 Suppose
Assumption
1
may have
that subjective probability distributions are given by Fg
and
diverging
information can increase disagreement.
that there exists
e
>
such that \B}
Then, there exists an open set of priors n^ and
[0, 1].
signals
We
lim
n—>(X)
n'^,
{p)
—
R'^ {p)\
such that for
all s
and Fq
>
e
that
for each
& S,
III
\4>i{s)-4>l{s)\>y-7r^\;
'
in particular,
Pr'(lim \cPi{s)-4>l[s)\>y-n^\)=l.
Proof.
that \R^ (p)
while \n^
^
Fix Fg and Fq and take
-
i?2 (p)|
'^'^\
—
^-
>
e for
each p e
tt^
=
[0, 1],
tt^
=
1/2.
lim„^c)o
\(t>n.
By Lemma
(s)
-
(p^ (s)|
>
Since both expressions are continuous in n^ and
neighborhood of 1/2 such that the above inequality uniformly holds
and the hypothesis
I
for
e'
tt^,
for
some
there
is
e'
>
0,
an open
each p whenever n^
Note, however, that the "no agreement to disagree" result derives from individuals' updating their beliefs
because those of others differ from their own (Geanakoplos and Polemarchakis, 1982), whereas here individuals
only update their beliefs by learning.
14
and n~ are
in this
and
signals, the
if
the original disagreement
is
(s
G 5)
=
1.
will interpret signals
relatively small, after almost all sequences of
disagreement between the two individuals grows. Consequently, the observation
common
of a
statement follows from the fact that Pr'
last
even a small difference in priors ensures that individuals
Intuitively,
differently,
neighborhood. The
sequence of signals causes an
initial difference
of opinion between individuals to
widen (instead of the standard merging of opinions under certainty). Theorem 4 also shows that
observing the same
both individuals are certain ex ante that their posteriors
will diverge after
sequence of signals, because they understand that they
will interpret the signals differently.
This strengthens our results further and shows that
for
some
priors individuals will "agree to
eventually disagree even more"
An
orems
interesting implication of
1
and
when
2,
there
Theorem
4
is
worth noting. As demonstrated by The-
also
learning under certainty individuals initially disagree, but each
is
individual also believes that they will eventually agree (and in fact, that they will converge to
his beliefs).
ically, let
This implies that each individual expects the other to "learn more". More specif-
1b=a be the indicator function
be a measure of learning
is
-
A^]
— A
for individual i,and let E'
Under
the probability measure Pr*).
so that E' [A'
for 9
= -
(tt'
-
tt^)^
certainty.
<
=
and A'
(tt*
—
—
Iq^a)
((f)^^
be the expectation of individual
Theorem
and thus E'
1
imphes that
W
<
[A']
[A^]
.
4>lo
=
'Poo
—
10=a)
(under
i
~
Ie=A,
Under uncertainty,
this
not necessarily true. In particular. Theorem 4 implies that, under the assumptions of the
theorem, there exists an open subset of the interval
this subset,
individual
we have
j.
W
The reason
also expects to learn
[A']
is
>
E'
[A-^]
,
[0, 1]
so that individual
that individual
i is
more from the sequence
will
tt^
are in
would expect to learn more than
initial
guess
tt',
but
of signals than individual j, because he believes
The
fact that
an individual
may
expect
play an important role in some of the applications in Section
4.
Non-monotonicity of the Likelihood Ratio
2.4
We
more than others
i
not only confident about his
that individual j has the "wrong model of the world."
to learn
such that whenever n^ and
next illustrate that the asymptotic likelihood ratio, R'
that
when an
(p),
may be non-monotone, meaning
individual observes a high frequency of signals taking the value
that the signals are biased towards a and
may put
have done with a lower frequency of a among the
15
a,
lower probability on state
signals.
he
A
may
conclude
than he would
This feature not only illustrates the
types of behavior that are possible
important
for the
when
individuals are learning under uncertainty but
apphcations we discuss in Section
also
is
4.
Inspection of expression (3) establishes the following:
Lemma
2 For any s € 5,
</)^ (s) is
decreasing at p (s)
Proof. This follows immediately from equation
When
FC
and only
if
B}
increasing at p
is
(s).
above.
(3)
non-monotone, even a small amount of uncertainty about the informativeness
is
may
of the signals
if
The next example
lead to significant differences in limit posteriors.
trates this point, while the second
example shows that there can be
illus-
"reversals" in individuals'
assessments, meaning that after observing a sequence "favorable" to state A, the individual
may have
a lower posterior about this state than his prior.
on asymptotic agreement
Example
and pb
1
be more systematically studied
will
(Non-monotonicity) Each
More
individual
p'
>
(1
precisely, for p'
>
(1 4-
are in a (^-neighborhood of
are not informative.
The impact
some
-I-
i
in the next subsection.
thinks that with probability
5) /2,
(5)
of small uncertainty
but with probability
/2, e
>
<
and 5
|j3^
>
e
0,
I
—
e,
pA
the signals
— p^|, we
have
P(„^i^+{^-^)/S ifpe{f-5/2,f + 6/2)
^^^^'
for
each 9 and
Now, by
i.
R'{p{s))
=
\
tfp{s)^{f-5/2,p^ +
i^^
^fp{s)e{l-p^-5/2^-p^ +
1
otherwise.
is
1
when
p{s)
is
(s) will also
0J,Q (s)
is
the
informative suggesting that the state
(/>to
(s) is small,
is
(s)
B, thus ^^^
informative, but this time favoring state A, so
uninformative and 4>^^[s)
falls
back to
tt*.
</>^ (s)
Intuitively,
16
=
(s)
goes back to
5/2)
and 5
>
to
1
1.
—
—
>
0.
p',
The
goes
likelihood
down
By Lemmas
1
to
and
1
2,
the signals are not informative, thus
In contrast, around
as the prior, 7r\
become uninformative again and
—
p\ and then jumps back
be non-monotone: when p
same
8/2)
small, takes a very high value at
afterwards, becomes nearly zero around
</'oo
for e
1
'
is
TTiffz^y
This and other relevant functions are plotted in Figure
ratio FC {p{s))
^
the asymptotic likelihood ratio
(4),
{
otherwise
e
7r\
^
1
0.
—
p',
become very
After this point, the signals
Around
1.
the signals
p',
the signals are again
Finally, signals again
when p{s)
is
around
1
—
become
p' or p', the
R'
4
Yl-'PI
<I>L
1
l-Tt'
71'
Tt^-7l'
1-7:2
7r'
4-^
i-p'
\-p'
Figure
1:
The
as
•kL-'
-1—+
P
P
three panels show, respectively, the approximate values of R} (p),
(f>^^,
and
0.
e
individual assigns very high probability to the true state, but outside of this region, he sticks
to his prior, concluding that the signals are not informative.
The
important observation
first
range of limiting frequencies, as
event that he will learn
as
e
^
and
either 1
— j5*
5
or
—
>
0,
^
e
This
9.
that even though
is
and
(5
—
equal to the prior for a large
is
(?!)^
each individual attaches probability
>
1
to the
because as illustrated by the discussion after Theorem
is
1,
each individual becomes convinced that the limiting frequencies will be
p^'.
However, asymptotic learning
individual also understands that since 5
region where he learns that 9
=
Each
considerably weaker than asymptotic agreement.
is
<
|p^
— p^|, when
the long-run frequency
is
in
a
A, the other individual will conclude that the signals are
uninformative and adhere to his prior
Consequently, he expects the posterior beliefs
belief.
of the other individual to be always far from his.
Put
differently, as
e
—
>
and
5
—
>
0,
each
individual believes that he will learn the value of 9 himself but that the other individual will
fail
to learn, thus attaches probability
the third panel of Figure
to learn,
1;.
at each
1
to the event that they disagree. This can be seen
sample path
and the difference between their limiting posteriors
following "objective"
TF^
=
1/3 and
one of the individuals
will
will fail
be uniformly higher than the
bound
mm {n\7T^,l-7r\l
When
in S, at least
from
tt^
—
2/3, this
bound
is
ttMttI
-1}-
equal to 1/3.
In fact, the belief of each indi-
vidual regarding potential disagreement can be greater than this; each individual believes
17
that he will learn but the other individual will
Pr' (lim„^oo
\(l>i
(s)
-
> Z) >
<Pl (s)|
I
-
where
e,
fail
as
e
to do so.
->
0,
Z
Consequently, for each
^ min {n^n^, 1-7t\1~-
i,
tt^}.
This "subjective" bound can be as high as 1/2.
The next example shows an even more extreme phenomenon, whereby a high frequency
s
=
a among the signals
Example
may
(Reversal)
2
reduce the individual's posterior that 9
Now
—A
below
of
his prior.
suppose that individuals' subjective probability densities are
given by
for
each 9 and
=
i
1,2,
r
(l-e-e2)/(5
ifp^-6/2<p<f + S/2
[
e^
otherwise
where
>
e
0, p'
>
<
and
1/2,
6
<
-
p^
p'^.
Clearly, as
e
—
0, (4)
>
gives:
tfp{s)<l-f-5/2,
orl-f + 5/2<pis)<l/2,
or f -5/2<p (s) <f + (5/2
ff(p(5))oo
otherwise.
Hence, the asymptotic posterior probability that 9
=
A'lS
ifp{s)<l-f-6/2,
or 1 - p' + S/2 < p{s) < 1/2,
orf -S/2 < p{s) <f- + 5/2
1
<t^oo{p{s))^{
otherwise.
Consequently, in this case observing a sufficiently high frequency of
posterior that 9
=A
below the
prior.
monotone, at
agree.
when
To
least
it
is
the state
is
crucial that the likelihood ratio R}
>
fP
Given the form of R^
A, p{s) will be close to
high probability to the event that 9
j
A
would
also converge to 1 as e
will learn the true state
and that
He
p'.
— A when
would assign even higher probability to
on
\4>]^
is
one of the individuals would expect that their
see this, take p*
—
>
=
a
may
Moreover, the individuals assign probability
there will be extreme asymptotic disagreement in the sense that
In both examples,
s
0.
yl
p
(p (s))
1
—
e
(?i^ (p (s))
not monotone.
If i?'
|
that
^
1.
were
beliefs will asymptotically
i is
almost certain that,
also understands that j
would assign a very
(s)
(p),
= p' >
at p (s)
=
p'
individual
p'. If
R^ were monotone, individual
and thus
Therefore, in this case
i
his probability
will
assessment
be almost certain that j
their beliefs will agree asymptotically.
18
—
reduce the
Theorem
1
established that there will be asymptotic agreement under certainty.
have thought that as
e
—+
and uncertainty disappears, the same conclusion would apply. In
show that even as each F^ converges to a Dirac distribution (that
contrast, the above examples
mass to a
assigns a unit
point), there
two individuals. Notably
and
p^
(5
—
—
+
p^
One might
this
is
may be
significant asymptotic disagreement
when
true not only
there
is
between the
negligible uncertainty,
but also when the individuals' subjective distributions are nearly identical,
0,
—+
This suggests that the result of asymptotic agreement in Theorem
.
—
e
i.e.,
i.e.,
may
1
>
as
not
be a continuous limit point of a more general model of learning under uncertainty.^ Instead,
we
next subsection that whether or not there
will see in the
is
asymptotic agreement under
becomes more and more concentrated around a point)
approximate certainty
(i.e.,
determined by the
properties of the family of distributions Fg
tail
Agreement
2.5
In this subsection,
is
Disagreement with Approximate Certainty
cind
we
as Fg
characterize the conditions under which "approximate certainty" ensures
asymptotic agreement. More
specifically,
we
will
study the behavior of asymptotic beliefs as
the subjective probability distribution Fg converges to a Dirac distribution and the uncertainty
As already
about the interpretation of the signals disappears.
Fq converges to a Dirac distribution, each individual
he
will learn
agreement
particular,
in the limit
on their
tail
as
is
considerably more
learning, this does not guarantee that the individuals will believe
that they will also agree on
totic
1,
become increasingly convinced that
However, because asymptotic agreement
the true state.
demanding than asymptotic
will
by Example
illustrated
We
6.
will
demonstrate that whether or not there
is
asymp-
depends on the family of distributions converging to certainty
properties. For
many
natural distributions, a small
tainty about the informativeness of the signals
is
amount
—in
of uncer-
sufficient to lead to significant differences in
posteriors.
To
state
and prove our main
density functions
that Fg.^
—^Fg^
there are
many
fg^
for
where
different
i
=
Fg^
ways
result in this case, consider
1,2, 9
£ {^,-B} and
assigns probability
in
1
m
to
p
a family of subjective probability
€ Z+, such that
as
= p'
e
for
some
p'
m^
cxd,
we have
(1/2, 1). Naturally,
which a family of subjective probability distributions
may
'Nevertheless, it is also not the case that asymptotic agreement under approximate certainty requires the
support of the distribution of each Fg to converge to a set as in Theorem 2 (that does not assign positive
probability to p\
<
1/2). See
Theorem
5 below.
19
converge to such a limiting distribution. Both for tractabihty and to make the analysis more
concrete,
we
focus on families of subjective probability distributions
We
a determining density function /.
(i)
(ii)
/
is
symmetric around
<
there exists x
<
/^
^^
impose the following conditions on
parameterized by
>
/:
zero;
oo such that / [x]
is
decreasing for
x>x\
all
(iii)
R{x,y)^
exists in
Conditions
oo] at all (x, y)
[0,
and
(i)
introduces the function
(ii)
—
(x
/m, which
y)
scale
e K^.^°
which
down
will arise naturally in the
the real line around y by the factor 1/m.
densities for individuals' beliefs about
transformation x
i-^
[x
—
p')
/m}^
each 9 and
i
fff^isa proper
hm„->oo
(Pn^m (^)
where
c'
(m)
individual
as
m—
>
is
i
^^^'
1/ Jq f
[m
[p
scaled
2 below).
The
of the form
dp
is
for
[0, 1]
fg^- In
x
i—
family of subjective
we consider the following family of
— p^))
(iii)
^^ determined by / and the
densities
(8)
a correction factor to ensure that
We
each m.
^ the limiting posterior distribution of individual
is
and
= d{m)f{m{p-f))
probability density function on
probability density of signals
about Pa
=
pA and pB, {fom}'
In particular,
fl^ip)
for
1
we consider mappings
of uncertainty,
Condition
study of asymptotic agreement
asymptotic statistics (see Definitions
in
amount
In order to vary the
(7)
are natural and serve to simplify the notation.
R (x, y),
and has a natural meaning
(j^
lim
m-»oo j [my)
i
also define
when he
(f)]^
^ =
believes that the
this family of subjective densities, the uncertainty
down by 1/m, and /^^ converges
to unit
becomes sure about the informativeness of the
mass
at p' as
1
oo, so that
signals in the limit. In other words,
oo, this family of subjective probability distributions leads to
(and ensures asymptotic learning; see the proof of Part
m^
of
Theorem
approximate certainty
5).
'° Convergence
will be uniform in most cases in view of the results discussed following Definition 1 below
(and of Egorov's Theorem, which links pointwise convergence of a family of functions to a limiting function to
uniform convergence, see, for example, Billingsley, 1995, Section 13).
'^This formulation assumes that p\ and
pg
are equal.
We
can easily assume these to be different, but do not
for such differences in the context
introduce this generality here to simplify the exposition. Theorem 8 allows
of the
more general model with multiple
states
and multiple
20
signals.
The next theorem
characterizes the class of determining functions / for which the resulting
famil}' of the subjective densities
s
/^
leads to asymptotic agreement under approximate
^jj [
certainty.
Theorem
5 Suppose that Assumption
subjective densities
(iii)
+ p^ —
1,1?^
1.
lim„^^oo
2.
Suppose that
rh G
"^e/zned in (8) for
fgmf
Suppose that f [mx)
above.
of {p^
I
— P^\)-
(</'j>o,m
R (p^
>
R
(^p^
0, there exists
=
—
—
1, \p^
^/ ^"^^
°^^y
1,2,
1/2, with
to
consider the family of
f satisfying conditions
(i)-
R{x, y) over a neighborhood
^ {P^ + P^ -
^f
p^\)
—
4>l,m
{s)\>e)<S
0.
Then for every
l^;,,^, (5)
+ p^ —
ttt.
-
l,\p^
e Z+ such
—
p'^\)
^0. Then
e
>
1,
- P^l) =
\f
and
>
S
0.
0, there exists
(Proof of Part
Hence, by
(5),
definition,
we
lim^^oo
there exists
e
=
>
1,2).
such that for each
>l-5
=
(Vm > m,2
1,2).
Let R!^ (p) be the asymptotic likelihood ratio as defined in
1)
One can
with subjective density fg ^.
(4) associated
(Vm > m,i
that:
Pr'flim \cl>l^is)-cf>l^{s)\>e)
Proof.
—
such that
Z-i-
Suppose that
5
>
p*
{my) uniformly converges
<PL,m, (P'))
+p'^
Pr' ( lim
3.
some
i
Then,
-
(P')
/f
For each
holds.
1
{(Plc,m (p')
- ^,m
(p')) "= ^
easily check that limm->oo
Rm
RL
(p')
if
and only
if
lim^^oo
(pO
=
0.
~
'^•
By
have:
m-^oo
m-^oo
fym-ypp))
= R{l-p'-p\f-p^)
= i?(pl+p2_l_|pl_p2Q^
where the
last equality follows
that lim,„-^oo Rln
(p')
=
by condition
the symmetry of the function /. This establishes
(i),
(and thus lim^^oo
('/''oo.m
{f)
-
</4>,m {?'))
=
0)
and only
if
if
.R(pi+p2-l,|pi-p2|) =0.
(Proof of Part 2) Take any
By Lemma
1,
there exists
e'
>
e
>
and 6 >
such that
0,
and assume that
</)Jx3.m
21
(P i^))
>
1
R [p^ + p^ —
— e whenever
1, \f?-
ff [p
(s))
—
<
p^|)
e'.
=
0.
There
also exists xo such that
/Xo
f{x)dx>l-
5.
(9)
Xo
Let
K,
=
min^-gr
^u^j;;,]
>
(x)
/
0.
Since / monotonically decreases to zero in the
<
above), there exists xi such tl>at / {x)
0.
Then,
for
any
m
> mi and
p{s) G
c'k
(p'
whenever
\x\
>
Let
\xi\.
mi =
— xo/m,p' + xo/m), we have
+ xi) /
(xq
—
\p (s)
tails (see (ii)
1
~^) >
(2p^
> xi/m,
+p*|
and hence
Therefore, for
all
m>
mi and
p{s) £
{p^
— xo/m,p' + xo/m), we
have that
0L,m(p(s))>l-e-
Lemma 1,
Again, by
e".
Now,
for
there exists e"
>
(10)
such that 4^^^^ iP
(*))
> ^~^ whenever
Bin {p (s))
<
each p{s),
\im B?^{p{s))
m—>oo
= R{p{s)+f-l,\p{s)-f\).
"
(11)
'
Moreover, by the uniform convergence assumption, there exists
uniformly converges to B[p{s)
+ fP -
— fp\) on
\,\p{s)
(p'
—
>
77
T],p^
such that Bin {p
(s))
+ r)) and
B{p{s)+p'-l,\p{s)-f\)<e"/2
for
each p
uous at
0).
—
(s) in (p'
(p^
+ p^ -
+t]). Moreover, uniform convergence also implies that
ri,p^
1, |p^
~ P^l)
<
Hence, there exists m,2
Bi, [p
Therefore, for
all
m
>
m,2
(s))
(ai^d in this
part of the proof, by hypothesis,
00 such that for
m > m.^
all
<B{p [s) +fP-l,
and p
[s]
£
(p'
—
\p
+
T],p^
and p
{s)-fP\)+
rj),
(s)
6
e"/2
(p'
<
—
it
B
is
contin-
takes the value
r],p'^
+ r]),
e".
we have
(t^oo,m{p{s))>l-e.
Set
(p'
m s
Then, by
max{7Tii,m2,77/xo}.
- xo/m,p' +xo/m), we have
Pr^ (|(^'^^„ ip{s))
-
establishes that Pr*
<jP^^^^
{p{s))\
{\<Plo,m.
(p
(•s))
(10)
iP (s))
-
<t\e^A) >
I
\<t>lc^m
-
<Pio,m (pi^))
22
|
and
(12)
(12),
for
< £
(pi^^m iP is))\
<
By
S.
e)
>
1
-
the
(^
any
m
Then,
symmetry
for
m
>
> m.
fh
and p{s) e
(9)
implies that
of
A
and B,
this
(Proof of Part 3) Since linim—oo-Rm
strictly positive, Vimm-^oo 4>io,m
<
(f)
= R[p^ ^p^ -
(p*)
We
1-
set
=
e
(l
-
1, \p^
~P^|)
limm-.oo
is
assumed
to be
/2 and use
<?!4),m (P*))
similar arguments to those in the proof of Part 2 to obtain the desired conclusion.
Theorem
complete characterization of the conditions under which approximate
5 provides a
certainty will lead to asymptotic agreement. In particular,
certainty ensures asymptotic learning,
may
it
agreement under
tainty
may be
The
full certainty.
Theorem
5,
shows
that, while
that there will always be asymptotic
1
shows that even a small amount of uncer-
instead,
cause disagreement between the individuals.
sufficient to
part of the theorem provides a simple condition on the
first
of the distribution
tail
/ that determines whether the asymptotic difference between the posteriors
approximate uncertainty. This condition can be expressed
R{p'+f-l,p'-p')^
The theorem shows
that
approximate
not be sufficient to guarantee asymptotic agree-
This contrasts with the result in Theorems
ment.
it
if
this condition
lim
\,
small under
as:
,./
-.2^^
-"•
13
then as uncertainty about the informa-
satisfied,
is
is
tiveness of the signals disappears the difference between the posteriors of the two individuals
will
become
negligible. Notice that condition (13)
Intuitively, condition (13)
individual will learn.
is
is
symmetric and does not depend on
related to the beliefs of one individual on whether the other
Under approximate
certainty,
we always have
that limm->oo R\n (p')
so that each agent believes that he will learn the value of 9 with probability
agreement
(or lack thereof)
the value of
When R
0.
+fP'~l,
|p^
~P^|)
frequency of p^ in the asymptotic distribution
frequency
be p^
when
,
p^
is
still
R(j)^
.
Therefore, individual
1,
who
=
0;
is
l,\p^
— fP'\) ^
0,
and thus expects to learn
R (p^ + J? —
—
\, \p^
learn the true state
p^l)
when
^0, an
individual
limiting frequency
the limiting frequency will be p^ (or
1
— p^),
0;
Asymptotic
will learn
when
the hmiting
almost certain that the limiting frequency will
individual
signals will be p^
—
an individual who expects a limiting
will still learn the true state
believes that individual 2 will reach the
+ f? —
1.
depends on whether he also believes the other individual
{p^
i.
1
same
still
is
himself.
who
inference as himself.
In contrast,
certain that limiting frequency of
However, he understands that, when
expects a limiting frequency of p^ will
happens to be
p^. Since he
is
fail
to
almost certain that
he expects the other agent not to learn the truth
and thus he expects the disagreement between them
to persist asymptotically.
Parts 2 and 3 of the theorem then exploit this result and the continuity of
23
R
to
show that
the individuals will attach probability
when
their beliefs will disappear
disagreement when (13)
mate
fails
to the eveift that the asymptotic difference between
1
(13) holds,
and they
will attach probability 1 to
Thus the behavior
to hold.
asymptotic
of asymptotic beliefs under approxi-
certainty are completely determined by condition (13).
Theorem
on whether
means
5 establishes t^at
R (p^ +p^ —
whether or not there
~
1, \p^
equal to
i^
P^])
determining distributions /.
for
will
We
0.
be asymptotic agreement depends
next investigate what this condition
Clearly, this will
depend on the
tail
behavior of /,
which, in turn, determines the behavior of the family of subjective densities ifg^}- Suppose
X
= p^ + p'^ —
1
>
p^
— p^ =
y
>
Then, condition (13) can be expressed as
0.
lim
m—>oo
=
41^
/ [my)
0.
This condition holds for distributions with exponential
On
normal distributions.
the other hand,
it fails
for distributions
example, consider the Pareto distribution, where / (x)
Then,
for
such as the exponential or the
tails,
is
with polynomial
proportional to |xp" for some
will
a negligible
amount
1.
be independent of
m
to converge even
fail
when
of uncertainty. In fact, for this distribution, the asymptotic beliefs
(since R\^ does not
=A
the asymptotic posterior probability oi d
,,:
(
(
depend on m).
according to
If
we take
tt^
=
=
tt^
1/2, then
is
i
(p(s)-p')""
\\-
+(p(s)+p'-l)
[p{s)-p')
for
a >
>o
This implies that for the Pareto distribution, individuals' beliefs will
is
For
each m,
fjmx) ^ (^\~°'
there
tails.
any m.
As
illustrated in Figure 2, in this case (p^^
the previous subsection explained
to breakdown).
To
case, (/>^„,(p(s))
see the
is
why
it
^
is
not monotone (in fact, the discussion in
had to be non-monotone
asymptotic agreement
for
magnitude of asymptotic disagreement, consider p
approximately
1,
and
<Pio,m (P i^)) is
(s)
= p'.
approximately y~"/ (x~"
In that
-f
Hence, both individuals believe that the difference between their asymptotic posteriors
j/~°).
will
be
™—
I'rcxj.m
This asymptotic difference
is
r'oo.Tnl
C-"
-I-
J/~"
increasing with the difference y
= p^ ~ P'^,
which corresponds to
the difference in the individuals' views on which frequencies of signals are most
24
likely.
It is
Figure
limn^oo
2:
4>n i^) fo^
Pareto distribution as a function of p
(s) \a
—
=
2, p^
3/4.]
^
also clear from this expression that this asymptotic difference will converge to zero as y
(i.e.,
—
as p^
p'^).
>
Proposition
D=
A
{{x,y)\
>
and
This
1 In
—
me
1
last
statement
Theorem
< X <
<
l,\y\
any such p^ and
assume
y} for some y
\p^
>
0.
—
p^\
>e
this proposition,
notation in that proof. Since
A
in addition,
lim
Pr'
X, there exists
indeed generally true
such that whenever
(0, cx))
Proof. To prove
5,
is
R
is
R
that
is
when
continuous:
>
e
and
>
5
(Vm > m,i =
<<5
of Part 2 of
continuous on the compact set
D
1,2).
Theorem
and
R {x, 0) =
p^.
Then, by the uniform convergence assumption, there exists
~P^|) < ^"/^ whenever
R (p (s) + p* —
1,
\p{s)
-
p'
|)
on
(p*
\p^
—
rj
r],p'^
each
for
~ P^\ <
such that
that R^i {p (s)) uniformly converges to
and use the
5
>
1, \p^
there exist
0,
A,
we modify the proof
R (p^ +p^ —
is
continuous on the set
Then for every
<
R
^-
>
Fix
such
+ rj)
and
Ri^p[s)+p' -l,\p{s)-fP\) <e"l2
for
each p
Theorem
(s) in (p'
—
ri,p^
+ tj). The
rest of the proof
is
identical to the proof of Part 2 in
5.
This proposition impHes that
of signals, then
if
the individuals are almost certain about the informativeness
any significant difference
in their
asymptotic beliefs must be due to a significant
difference in their subjective densities regarding the signal distribution
that
when
\p^
p^
— p^l
=
p^,
is
not small). In particular, the continuity of
we must have
R
(p^
+p^ —
1, \p^
25
—
p'^j)
—
0,
R
in
(i.e., it
must be the case
Proposition
1
implies that
and thus, from Theorem
5,
there
pi
significant differences in asymptotic beliefs. Notably, however, the requirement that
be no
will
=
p2
be asymptotic agreement
for all p^jP^
>
1
established that mider certainty there will
1/2.
R
worth noting that the assumption that
It is also
relevant range
is
important
illustrated a situation in
bounded away from
We
Theorem
rather strong. For example.
jg
for the results in Proposition
which
this
assumption
next focus on the case where p^
^
and the asymptotic
failed
Example
differences
1
remained
p'^.
agreement under approximate certainty.
first define:
A
Definition 1
density function f has regularly- varying tails if
satisfies
l^ =
Mm
>
for any x
The
condition in Definition
from the fact that
H{x)H{y)
1
that
=
it
H (x)
is
only possible
any compact subset of (0,oo).
tails,
is
implies that
in the limit, the function
H{xy), which
M
e
H
(•)
has unbounded support and
:
if
H{x)
H{x)
1
=
x""' for
This implies that
we have
assumption) are
satisfied.
same expression
as for the Pareto distribution,
= x""
for
but nevertheless has
a £
(0, oo).
a G
This follows
Moreover, Seneta
(0,00).-^^
holds locally uniformly,
then the assumptions imposed in Theorem 5
In fact,
relatively weak,
must be a solution to the functional equation
(1976) shows that the convergence in Definition
in
H{x) e
it
0.
important implications. In particular,
X
continuous in the
p^ and provide a further characterization of which
classes of determining functions lead to asymptotic
We
(p) ^^
In particular, recall that
1.
gap between p^ and
zero, irrespective of the
Rm
or lirn^^o
if
i.e.,
uniformly for
a density / has regularly-varying
(in particular,
that, in this case,
R
the uniform convergence
defined in (7)
is
given by the
X
R{x.y)
\y
and
is
everywhere continuous. As this expression suggests, densities with regularly- varying
behave approximately
varying
.
tails
'"To see
this,
like
can be written as /(x)
Haan
=
in the tails; indeed a density
£(x)x~"
note that since limm^oo {f{mx)/f{m))
//(.y)=
See de
power functions
(lp^)=
lim
^-^00 \
f(m) J
for
= H (x)
lim
26
(x)
with regularly-
some slowly-varying function C (with
£ R, we have
ffe)^) =//(.)
Tn-.^\f(my) f {m) J
(1970) or Feller (1971).
/
tails
ff(,)
=
limm->oo 'C(mx)/£ (m)
and
1).
Many common
distributions, including the Pareto, log-normal,
t-distributions, have regularly-varying densities.
Definition 2
A
We
also define:
density function f has rapidly- varying tails
x>l
if
4t^
lim
" " / (m)
^
^
-
x-°°
=
<
As
>
if
oo
if
'
x =
^
X <
,
.
•'
for any x
1
if it satisfies
l
,
I
0.
above convergence holds locally uniformly (uniformly
in Definition 1, the
compact subset that excludes
Examples
1).
x over any
in
of densities with rapidly- varying tails include the
exponential and the normal densities.
Prom
totic
these definitions, the following corollary to
agreement under approximate certainty to the
Theorem
tail
5
is
immediate and
links
asymp-
behavior of the determining density
function.
Corollary
1.
1 Suppose that
1
holds
and p^ y^P^-
Suppose that in Theorem 5 f has regularly-varying
that for each 5
>
0, there exists rh
Pr'fhm
2.
Assumption
S
W
(s)|>e)
|(^^,„(s)-</.2
varying
tails
there exists
(Vm > m,i =
>1_<5
e
>
such
tails.
Then for every
e
1, 2).
>
and 5 >
0,
1^+ such that
(
lim
|</.^,„ (s)
-
<„ {s)\>e)<5
This corollary therefore implies that whether there
on whether the family
Then
^J^j^ such that
Suppose that in Theorem 5 f has rapidly-varying
there exists fh
tails.
will
(Vm > m,i =
1,2).
be asymptotic agreement depends
of subjective densities converging to "certainty" has regularly or rapidly-
(provided thatp-^
^
p^).
Returning to the intuition above. Corollary
that the failure of asymptotic agreement
about limiting frequencies,
i.e.,
p^
^ fP,
is
1
and the previous definitions make
it
clear
related to disagreement between the two individuals
together with sufficiently thick
probability distribution so that an individual
who
27
tails of
the subjective
expects p^ should have sufficient uncertainty
when
confronted with a Umiting frequency of p^
this
sufficient for
is
both individuals to believe that they
but that the other individual
become
Along the
.
do
will fail to
lines of the intuition given there,
will learn the true value of Q themselves,
Rapidly-varying
so.
imply that individuals
tails
model of the world and thus when individual
relatively certain of their
by sampling
limiting frequency p close to jDut different from p', he will interpret this as driven
~
variation and attach a high probability io 9
A. This will guarantee asymptotic agreement
between the two individuals. In contrast, with regularly-varying
from
certainty, limiting frequencies different
but as potential evidence
for 6
=
The previous
be interpreted not as a sampling variation,
B, preventing asymptotic agreement.
section provided our
signals. In this section,
and L >
states
K
To
=
signals.
The main
an environment with two states and two
an environment with
results generalize to
e
and
results parallel those of Section 2,
all
K>2
the proofs for
Appendix.
Q=
{A^, ...,A^^
is
a set
K >2 distinct elements. We refer to a generic element of the set by A'^. Similarly,
...,a^}, with L > K signal values. As before, define s = {st}^^, and for each
{a-^,
1, ...,L, let
r^(s)
be the number of times the signal
large
numbers imphes
surely converges to
pis)
=
the set
p' (s)
G
=
[0, 1]
(pi(s),...,p^(s)), where
a'
out of
n
first
a'|
both individuals,
with YliLi
A(L) = {p -
~
P^ i^)
(pi,
.
.
.
Once
signals.
for
^-
each
I
again, the strong law of
=
1,
Define p{s) e
,p^) e
[0,
1]^
L, r^ (s) /n almost
...,
A (L)
as the vector
Eti p' =
:
1
}
.
and
let
S be
With analogy
individual
a}
St
= #{i<n|si =
that, according to
some
5=
—
results in
generalize our results to this environment, let 9 E Q, where
containing
let St
main
we show that our main
this section are contained in the
s
p' will
even under approximate
tails,
Generalizations
3
/
observes a
i
i
<
s
:
lim„_>oo
?"i
to the two-state-two-signal
assigns to ^
when
6 5
=
A'', tt'
the true state
is
9.
=
(s)
/n
model
exists for each
immediate generalizations of Theorems
1
=
in Section 2, let 7r|
(ttj, ...,7r^),
When
I
1,
...,
>
L
(14)
\
be the prior probability
and pg be the frequency
of observing signal
players are certain about pg's as in usual models,
and 2 apply. With analogy to
the joint subjective probability distribution of conditional frequencies p
28
before,
=
we
define Fg as
(pg, ...,p^)
according
to individual
to
Since our focus
i.
Assumption
Assumption
and
is
2 For each
and
i
the distribution Fq over
0,
=
also define (pl^ni^)
probability that 6
—
P^* [9
—
lines discussed in
{st}"=o) ^°^ ^^^^ ^
A'^
\
4,oo(p(s))= lim
n—>oo
'
it
is
Remark
~
2 above.
l,...,K as the posterior
observing the sequence of signals {st}^^Q, and
A'' after
this structure,
A(L) has a continuous, non-zero
A(L).
This assumption can be weakened along the
Given
similar
1.
finite density pg over
We
we impose an assumption
learning under uncertainty,
4>l^nis)'
straightforward to generalize the results in Section
define the transformation T^
:
2.
Let us
now
M.^ -^ M.^~^, such that
Tk{x)= (^,k'e{l,...,K}\k
Here T^
(x)
is
taken as a column vector.
theorems and the proofs. In particular,
This transformation
this transformation will
will play
a useful role in the
be apphed to the vector
priors to determine the ratio of priors assigned the different states by individual
define the
norm
||x||
The next lemma
Lemma
—
max;
|2;|'
generalizes
for
x
—
Lemma
(x^,
.
.
.
i.
tt^
of
Let us also
,x^) G M^.
1:
3 Suppose Assumption 2 holds. Then for
all s
G 5,
Ek'^k^lfLApi.^))
1
Our
first
theorem
Theorem
in this section parallels
2 there will be lack of asymptotic learning,
and under a
3
and shows that under Assumption
relatively
weak additional
condition,
there will also asymptotic disagreement.
Theorem
1.
6 Suppose Assumption 2 holds fori
VT'{<t^,^{p{s))^l\e
=
A^)
=
=
l,and
29
1,2,
then for each k
—
1,
.-.^K,
and for each
2.
Pr^ {\<i>l^ (pis))
0)
The
0)
=
0,
=
and F^
-
^
<Pl^ ipis))\
= F^
for each
6*
=
0)
1
wheneverPi^in {n')-n
{ir''))'n{f{p{s))
=
W{{Tk
{TT'^))'Tk{f{p{s))
=
e 0.
Theorem
additional condition in part 2 of
plays the role of differences in priors in
of the vector in question). In particular,
relative asymptotic hkelihood of
some
6,
that
Theorem
[iv^)
3 (here "
-Tk
'
"
denotes the transpose
then at some p(s), the
this condition did not hold,
if
states could
be the same according to two individuals
with different priors and they would interpret at least some sequences of signals in a similar
manner and achieve asymptotic agreement.
Pr^((Tfc (tt^)
it
-
Tfc (7r2))'Tfc(/'(/9(s))
=
=
0)
did not hold, a small perturbation of
is
or
tt^
important to note that the condition that
It is
tt^
weak and holds generically— i.e.,
relatively
would restore
The Part
it.^*^
2 of
Theorem
if
6
therefore implies that asymptotic disagreement occmrs generically.
The next theorem shows
that small differences in priors can again widen after observing
the same sequence of signals.
Theorem
each p G
7 Under Assumption
—
each k
[0, 1],
vectors n^ and
tt-^,
\<Pk,oo
1,
2,
,K,
assume
1'
=
where 1
[Tk
(1,
...,
-
Ufg
(p))^g0]
1)'.
Then, there exists an open set of prior
Tk
f
(/| (p))^gej j
¥=
for
such that
iP (s)) -9^fc,oo(p(s))|
>
kfc-7r|| for eachk
=
l,...,K and s e
S
and
Pr'
The
(kioo
condition
1'
(P (s))
(
condition in part 2 of
Tfc
(
-
'/'loo
(P(s))|
(/g ip))g^Q)
Theorem
6,
generically. Finally, the following
—
> KJ -
Tk
(
7r||)
=
1
(/| {p))g^Q)
)
7^
and as with that condition,
is
it is
theorem generalizes Theorem
tion of the families of probability densities
is
=
for each k
5.
1,
...,
K.
similar to the additional
relatively
weak and holds
The appropriate
construc-
also provided in the theorem.
'^More formally, the set of solutions S = {(7^^7r^p) € l\{Lf
{Tk (tt^) - Tk (7r^))'Tfc(/'(p)) = 0} has
Lebesgue measure 0. This is a consequence of the Preimage Theorem and Sard's Theorem in differential
topology (see, for example, Guillemin and Pollack, 1974, pp. 21 and 39). The Preimage Theorem implies that
— y, then f~^{y) is a submanifold of
if y is a regular value of a map /
with dimension equal to
dimX — dim Y. In our context, this implies that if is a regular value of the map {Tk (tt') —Tk (7i""))'Tfc(/'(p)),
then the set S is a two dimensional submanifold of A(L)^ and thus has Lebesgue measure 0. Sard's theorem
:
:
implies that
is
X
X
generically a regular value.
30
Theorem
8 Suppose that Assumption 2 holds.
subjective density fg
^
ever 9
^
9'
,
=
'^/
and f
Jp^^in f
M^ —
:
Q
and
m
6
Z_|_,
define the
by
= cii,e,m)f{mip-p{i,e)))
fi^{p)
where c{i,9,m)
For each 6 E
('7^
]R is
>
(p
- p{i,6)))
G
dp, p{i,6)
A (L)
(16)
withp{i,9)
^p
when-
{i,0')
a positive, continuous probability density function that satisfies
the following conditions:
(i)
limft^oo
niax|3..||^||>/j}
(x)
/
=
0,
R{x,y)^
(17)
and
exists at all x,y,
(Hi)
^^^
lim
m^oo / (my)
convergence in (1 7) holds uniformly over a neighborhood of each
{p{i,9)-p{j,9'),p{i,9)-p{j,9)).
Also
let 4>k^oo,m
(P i^))
subjective density fg
1.
^
^^ ^^^
asymptotic posterior of individual
i
with
Then,
.
lim^^oo (fk,oo,m
— l™n^oo 4>knm i^)
{P ih
A^))
-
^fc,oo,m {p (^'^'')) j
=
^ «/
""^ Only
if
R(p{i,A^)-p(^j,A'''yp{i,A'')-p{j,A''))^Oforeachk'y^k.
2.
Suppose that
for every
e
R [p
>
{i,
9)
and 5 >
—p
(j, 9')
— p (j, 9)) =
,p{i,9)
m
0, there exists
G
Z+
—p
Suppose that
R [p
(i,
there exists e
>
such that for each 5
9)
Pr^ (l|0L,m(5)
(j, 0')
-
,p{i,9)
>
<^^,^(5)1|
(Vm>7n,2 =
These theorems therefore show that the
— p {j, 9)) ^
0, there exists
>
e)
>
results
9'
.
Then
such that
W[\\4>l^,m{s)-4L,m{s)\\>e)<5
3.
for each distinct 9 and
1
-
<5
l,2).
for each distinct 9 and
m ^1.+
9'
.
Then
such that
iym>m,i =
1,2).
about lack of asymptotic learning and
asymptotic agreement derived in the previous section do not depend on the assumption that
31
there are only two states and binary signals.
1
and Corollary
The
1
It is also
to the case with multiple states
straightforward to generalize Proposition
and
signals;
results in this section are stated for the case in
and
states are finite.
and
states,
They can
we omit
this to avoid repetition.
which both the number of signal values
also be generalized to the case of a
continuum
of signal values
but this introduces a range of technical issues that are not central to our focus
here.
Applications
4
In this section
we
are chosen to
show various
discuss a
number
different
under uncertainty. Throughout, we
illustrates
how
application and shows
how
fourth example illustrates
in bargaining.
Damme
how
arise
(1993).
some simple
The
third
example
is
the most substantial
The
example shows how a special case of our model of learning
when
there
is
information transmission by a potentially biased
4.1
Value of Information in Common-Interest
Consider a common-interest game
interest
from basic game
learning under uncertainty can affect the timing of agreement
outlet.^"*
common
example
learning can act as an equilibrium selection
media
cally in
insights
first
learning under uncertainty affects speculative asset trading.
Finally, the last
under uncertainty can
The
strive to choose the simplest examples.
learning under uncertainty can overturn
device as in Carlsson and van
applications
economic consequences from learning and disagreement
The second example shows how such
theory.
The
of applications of the results derived so far.
in
Games
which the players have identical payoff functions. Typi-
games information
is
valuable in the sense that with
about underlying parameters, the value of the game in the best equihbrium
would therefore expect players to
more information
will
We
be higher.
collect or at least wait for the arrival of additional informa-
'^In this section, except for the example on equihbrium selection
manipulation, we study complete-information games with possibly
and
belief structure in these
n <
oo, consider the information partition /"
games can be described
=
{/"
as follows.
(s)
=
and the
last
non-common
example of the game of
priors.
Fix the state space
{{d,s')
\s[
=
st^t
< n}
|s
belief
Formally, information
Q =Q
x S, and
€ 5} that
is
for
each
common
for
both players. For n = oo, we introduce the common information partition /°° = {/°° (s) = 9 x {s} \s e S}.
At each /" (s), player i = 1,2 assigns probability (f>l^ (s) to the state 9 = A and probability 1 — 05, (s) to the
sate 6 = B. Since the players have a common partition at each s and n, their beliefs are common knowledge.
Notice that, under certainty, ip^^ (s) = 0^ (s) e {0, 1}, so that after observing s, both players assign probabihty
1 to the same 6. In that case, there will be common certainty of 9, or loosely speaking, 9 becomes "common
knowledge." This is not necessarily the case under uncertainty.
32
We now
tion before playing such games.
show that when there
is
learning under uncertainty,
additional information can be harmful in common-interest games, and thus the agents
game
prefer to play the
To
before additional information arrives.
matrix
illustrate these issues, consider the payoff
a
a
where
=
^
G
6'
1 is
TT
(3
29,29
1/2,1/2
1/2,1/2
1-9,1-9
and the agents have a common prior on 9 according to which probability of
{0, 1},
e (1/2,
1).
When
there
value of the payoff from (a, a)
from
(/3,/3) is strictly less
is
no information, a
dominates
than 1/2). In the dominant-strategy equilibrium,
First, consider the implications of learning
=
9\9)
=
p'
>
the agents will learn
will learn that
a,
and hence
P and
(a, a)
Theorem
1/2.
9.
9—1;
If
1
=
the frequency p
(s)
=
0.
the dominant strategy equilibrium.
=
if
1.
greater than 1/2, they
1 is
Up (s) <
1/2,
P
strictly
dominates
>
1/2,
a
strictly
dominates
If
p{s)
Consequently,
when they
certainty before playing the game, the expected payoff to each player
This implies that,
each player
{st}^j, where each agent believes that
of signal with sj
the dominant strategy equilibrium.
is
>
{a, a),
then implies that after observing the sequence of signals,
otherwise they will learn that 9
(/3,/3) is
(since the expected
under certainty. Suppose that the agents are
allowed to observe an infinite sequence of signals s
[st
/?
than 1/2 and the expected value of the payoff
strictly greater
is
strictly
receives 29 with probability n, thus achieve an expected payoff of 27r
Pr'
may
is 27r
-|-
(1
learn under
—
tt)
>
27r.
they have the option, the players would prefer to wait for the arrival of
public information before playing the game.
Let us next turn to learning under uncertainty. In particular, suppose that the agents do
not
know the
signal distribution
and
their subjective densities are similar to those in
Example
2:
(l
-
e
-
e^)
^
for
each
9,
where
the limit where
e
<
-^
6
(5
—
>
if
f -5/2<p<f + 5/2
(18)
otherwise
< p^—p^ and
and
/5
ifp<l/2
e
0,
e
and
6 are taken to
be arbitrarily small
or loosely speaking, where
33
e
=
and
6
=
(i.e.,
we consider
0).
Recall from
Example 2 that when
e
=
and 5
=
the asymptotic posterior probabihty of ^
0,
a
1
'PLiPis))
< l-p'-5/2,
+ 6/2 < p{s) <
p{s)
-p'
or
1
or
f- 5/2 <p{s) <f + 6/2,
=
1 is
1/2,
otherwise.
As discussed above, when
value of
9,
e
^
and 6
=
each agent beheves that he will learn the true
0,
while the other agent will reach the opposite conclusion.
agents expect that one of
them
will
have
(/>^
Consequently, the unique equilibrium will be (a,
payoff of 1/2, which
is
strictly less
arrival of information (which
agents
may
is 27r).
prefer to play the
game
and van
Damme
may be due
(1993),
we now
4>]^
ip{s))
=
0.
giving both agents an ex ante expected
there
is
learning under uncertainty, both
Games
initial difference in players' beliefs
it
while the other has
before the arrival of public information.
The
prior;
when
Therefore,
Selection in Coordination
common
/3),
1
than the expected payoff to playing the game before the
4.2
of
—
ip{s))
This implies that both
about the signal distribution need not be due to lack
on an example by Carlsson
to private information. Building
illustrate that
distribution, small differences in beliefs,
on the outcome of the game and may
when
the players are uncertain about the signal
combined with learning, may have a
select
significant effect
one of the multiple equilibria of the game.
Consider a game with the payoff matrix
_I
where 9
~
A/" (0,1).
where
G
{0, 1}
St
The
77
9,9
9-1,0
N
0,9-1
0,0
players observe an infinite sequence of public signals s
~ A/'(0, 1).
=
l\9)
=
l/{l
+ exp{-i9 +
Ui
is
{st}^o>
rj))),
+ Ui
uniformly distributed on [— e/2, e/2] for some small
Let us define k
(19)
In addition, each player observes a private signal
Xi=r]
where
=
and
PTist
with
N
I
=
\og{p{s))
the players infer that 9
+
t]
=
k.
—
log(l
—
e
>
0.
p{s)). Equation (19) implies that after observing
For small
e,
conditional on Xi,
34
77
is
s,
distributed approximately
uniformly on
[xi
and
ditional on Xi
Now
—
e/2,Xi
9
s,
is
+ e/2]
approximately uniformly distributed on
note that with the reverse order on
extremal rationalizable strategy
Nash
tion
/
Xj
if
<
X* and
x,
A'^ if
between the two actions,
O (e)
where
is
This
x*.
=
such that lime_<o
when
<
K-
1/2 and
—
Xi
—
e/2,
k—
+
Xi
e/2].
supermodular. Therefore, there exist
is
monotone, symmetric Bayesian
also constitute
a cutoff value, x*, such that the equilibrium ac-
cutoff, x*,
small, the
N
Xi
>
k
+
>
Pr(xj
O (e) =
e is
if
is
game
[k
defined such that player
is
i is
indifferent
i.e.,
X*
Therefore,
the
which
profiles,
>
K.-X*
Xi
Xi,
Equilibria. In each equilibrium, there
is
and van Damme, 1993). This implies that con-
(see Carlsson
game
=
X*)
=
1/2
+
(e)
This establishes that
0.
=
x*\xi
-
AC
is
1/2
- O (e)
dominance
.
solvable,
and each player
i
plays /
if
1/2.
In this game, learning under certainty has very different implications from those above.
Suppose instead that the players knew the conditional signal distribution
so that
we
common
are in a world of learning under certainty.
Then
after s
how
they knew
observed, 9 would
is
knowledge, and there would be multiple equilibria whenever 6 G
therefore illustrates
(i.e.,
(0, 1).
rj),
become
This example
learning under uncertainty can lead to the selection of one of the
equilibria in a coordination game.
A
4.3
One
Simple Model of Asset Trade
of the
trading.
most interesting applications of the ideas developed here
Models of
assets trading with different priors have
Harrison and Kreps (1978) and Morris (1996).
"speculative asset trading"
for the
.
We now
They
models of asset
by,
among
different priors
others,
about
establish the possibility of
investigate the implications of learning under uncertainty
pattern of speculative asset trading.
Consider an asset that pays
2
been studied
These works assume
the dividend process and allow for learning under certainty.
to
is
owns the
asset,
but Agent
1
1 if
the state
may wish
to
is
buy
A
it.
and
We
if
the state
is
B. Assume that Agent
have two dates, r
=
the agents observe a sequence of signals between these dates. For simplicity,
to be an infinite sequence s
=
{st}'^-^-
We
also simplify this
35
and t
—
I,
and
we again take
this
example by assuming that Agent
1
has
the bargaining power: at either date,
all
if
he wants to buy the asset, Agent
a take-it-or-leave-it price offer P^, and trade occurs at price Pr
Assume
also that
Agent
would
1
>
tt^
like to
tt^,
so that Agent
purchase the
game.
this
We
asset.
more
1 is
1,
Pa
both agents recognize
same
both of them:
for
at each history p{s).
1
— Pb =
at r =
it
is
that at r
>
p*
In that case, from
1/2.
=
1,
for
each p
,0
(s)
>
1/2 and
1 if
Therefore,
and the continuation value of Agent
is 0,
Agent
0,
2 accepts
offer the price
Pq
=
'""^
not
offer
if
p
(s)
—
know
p.4
and pB
for
and only
=
Agent
Agent
approximately
1 is
1
To
1,
=
n'^,
1.
0, his
with
ir^
=
4>to
=
1
(P i^)))
the continuation value of
0,
they trade at price Pq, then the
If
Since
.
tt^
>
tt'^,
Agent
1 is
happy
to
Therefore, with learning under
place.
0.
e
^
0).
trading, which
—
0.
>
is
Now,
continuation value
1
=
<^^ {p (s))
1
—
1
is
tt'^,
p^
—
5/2
and the value
<
p
(s)
<
of the asset
2 at
Since the equilibrium payoff
shows that when they do not
at least
(1
-
TT^)
value of Agent 2 must be at least
sum
1, if
buys the asset from Agent
tt^.
Therefore, they can trade at date
7r\ exceeds the
asset at date
at date
in all other contingencies, this
only
tt^,
if
of these continuation values, n^ (l
impossible, there will be no trade at r
buy the
e
now have
consider a simple example where
first
Hence, at such p{s), Agent
The continuation
is
we
enjoying gains from trade equal to
of never selling his asset.
to
Pq >
Agent 2
Tfl
is
=
[p (s))
(j)]^
r
Pq and Pq, respectively. This imphes that at
illustrate this,
Example
must be non- negative
trade at date
(when
if
—
and trade takes
5/2, then the value of the asset for
price Pi (p(s))
of
tt^
is tt^.
1/2. Hence, at
In contrast to the case of learning under certainty, the agents
subjective densities are as in
+
=
<
next turn to the case of learning under uncertainty and suppose that the agents do
an incentive to delay trading.
p^
if
at date r
be
2 will
be immediate trade at r
certainty, there will
We
an
and
1
2
Theorem
the value of the asset will the
(s),
trade does not occur at r
if
Suppose that each
learning under certainty.
some number
P^ for
be worth
will
continuation value of agents
date
assumption ensures that
optimistic. This
the agents will be indifferent between trading the asset (at price Pi
Agent
offer.
^
certain that
is
Agent 2 accepts the
if
are interested in subgame-perfect equilibrium of
Let us start with the case in which there
agent
makes
1
=
(provided that p
0.
(s)
Instead,
Agent
1 will
as he has the option
the total payoff from
—
tt'^)
+ tt^.
Since this
wait for the information
turns out to be in a range where he concludes
36
that the asset pays
1).
This example exploits the general intuition discussed after Theorem
uncertain about the informativeness of the signals, each agent
may
4:
if
the agents are
expect to learn more from
the signals than the other agent. In fact, this example has the extreme feature whereby each
agent believes that he will definitely learn the true state, but the other agent will
This induces the agents to wait
so.
towards
common
and make trades
beliefs
learning under certainty and
.trade at the
less likely.
the reason
is
common
why
do
to
information before trading.
for the arrival of additional
This contrasts with the intuition that observation of
fail
information should take agents
This intuition
is
correct in models of
previous models have generated speculative
beginning (Harrison and Kreps, 1978, and Morris, 1996). Instead, here there
is
delayed speculative trading.
The next
result characterizes the conditions for delayed asset trading
Proposition 2 In any subgame-perfect
E2[,/,y
That
is,
when
'PIo (p('S))
>
>
n'^
E^ [min
(plc (p(s))>'
tt^
< E^
Agent
his valuation of the asset
take place at the price Pq
At date
1,
Agent
max{(/)^ {p{s))
only
1
—
=
n'^,
[min {(^^, f/)^}], Agent
(j?^
is
Pr^ {6
=A
while trade at r
buys the asset
if
—
delayed to t
does not buy at t
1
Proof. In any subgame-perfect equilibrium. Agent 2
and hence
is
generally:
1 if
and only
if
=7r2>Ei[min{^^,0^}].
{<?^^,(/'oo}]'
when
equilibrium, trade
more
and only
=
|
is
=
and buys
buys at t
1
indiiTerent
=
at
if (p]^
will
between trading and not,
be at the price Pi (p
{p (s))
{p{s)) ,0}. This implies that Agent
> 0^
1 is
1 if
0.
Information). Therefore, trade at r
1
=
t
(s))
=
— 0^
can
{p (s)).
{p (s)), yielding the payoff of
willing to
buy
at r
=
if
and
if
TT^-TT^
>
Ei[max{(/.i,(p(s))-,^^(p(s)),0}]
=
B^
[</)L
=
tt'
-E'
(P (s))
- min
{4>l, {p is))
,
4>l {p is))}]
[min {(Plip is)), 4>lip{s))}],
as claimed.
Since n^
~
E^
[<?^io]
difference in beliefs,
tt^
— ^^ [min
—
tt^,
in
{(/>^, (fto]\
>
^^^^ result provides
terms of the differences
37
a cutoff value
for
the initial
in the agents' interpretation of the
signals.
The
cutoff value
[max
is E-^
- 0^
[cf)^ {p (s))
(p (s))
=
lower than this value, then agents will wait until r
,
O}]
the
If
.
initial difference is
to trade; otherwise, they will trade
1
immediately. Consistent with the above example, delay in trading becomes more likely
the agents interpret the signals more differently, which
This reasoning' also suggests that
cutoff value.
if
evident from the expression for the
is
=
Fg
Fg
each 6 (so that the agents
for
The next
interpret the signals in a similar fashion), ^^ then trade should occur immediately.
lemma shows
when
that each agent believes that additional information will bring the other agent's
expectations closer to his
own and
will
=
be used to prove that Fg
Fg indeed implies immediate
trading.
Lemma
4
>
If n^
tt^
=
and Fg
Fg for each
El
then
9,
> n\
[4>l]
Proof. Recall that ex ante expectation of individual
=
E^C,]
+
f\-K^f\{p)<i>>^{p)
i
regarding
(j>'^
can be written as
{l-i:')fh{l-p)4>^oo{p)]dp
(20)
Jo
~
where the
first line
f^ n^f^{p)
^^fAip)
Jo
+ {l-n^)fs{l-p)
=
fe (P)
=
Now
fe (p) ^°^ ^11 p.
r(^\^^^'=
f
(20),
E^
[(p^,]
increasing in n.
=
I {n^)
^/-4 (P)
and
tt^
and
+
(1
=
-
^) /b
fact that since
=
Fg
Fg,
=
E^ [0^]
(1
- P) .
f.,dp.
- p)\JA [P)
2\ t
l^
TT^) /s
(1
/
(tt^).
^^^^^
'Recall from
and the
(4)
Hence,
it
suffices to
show that /
is
Now,
JO
Moreover, /^
^^
define
^2t
(r^^l^
^
JA [P) + (1
I
Jo
From
^
uses the definition of ex ante expectation under the probability measure
Pr', while the second line exploits equations (3)
fe (P)
^
+ il-n^)fBil-py^^'^'^''
r2/.4(p)
TT-^
(p) / [tt^Ja (p)
Theorem
+
(l
3 that even
{fA{p)-fB{l-p))dp.
+ (l-7r2)/B(l-p)
-
tt^)
when Fq
/b
=
(1
-
p)]
>
1
if
and only
if
/^
(p)
> /b
Fg, agents interpret signals differently because
38
tt'
(1
^
-
tt'.
p).
Hence,
I'i^)
fA{p)
2f ( \^(C'^\f
n P)-AfA{p)-fB{l-p))dp
{^-T^-)
fBi.1JjA>fB'^ fA[p) +
=
[
-/
2, (r.^^!^^^''\^f n
JfA<fB ^ /^ yP) + (1 - ^ ) JB (1
>
.
Ub{1- p)-
fA[p))dp
P)
UB{l-p)-fA{p))dp
UA{p)-fB{l-p))dp- f
[
JfA<fB
JfA>fB
=
-
/ ifA{p)-fBil-p))dp^O.
Jo
Together with the previous proposition, this lemma yields the following result establishing
that delay in asset trading can only occur
when
subjective probability distributions differ across
individuals.
Proposition 3
atT
=
//
Fq
4.
Fq for each
9,
then
m
any subgame-perfect equilibrium, trade occurs
0.
Proof. Since n^
each p
=
(s).
>
iP'
Then, E^ [min
and
{<?!>^,
Therefore, by Proposition
Fl}
—
(pto}]
2,
Agent
i?^,
~
1
E^
Lemma
[^to]
-
buys at r
1
imphes that (l)^{p{s)) > 4>1oip{s))
""^i
=
where the
last inequality is
by
for
Lemma
0.
This proposition establishes that when the two agents have the same subjective probability
distributions, there will be
when Fg ^ Fg, delayed
when
no delay
in trading.
speculative trading
is
However, as the example above
possible.
The
intuition
is
given by
illustrates,
Lemma
4:
agents have the same subjective probability distribution but different priors, each will
believe that additional information will bring the other agent's beliefs closer to his own. This
leads to early trading. However,
when the
distributions, they expect to learn
Theorem
agents differ in terms of their subjective probability
more from new information (because,
as discussed after
4 above, they believe that they have the "correct model of the world"). Consequently,
they delay trading.
Learning under uncertainty does not necessarily lead to additional delay
actions, however.
Whether
it
in
economic trans-
does so or not depends on the effect of the extent of disagreement
on the timing of economic transactions.
We
the presence of learning under uncertainty
will
next see that, in the context of bargaining,
may be
a force towards immediate agreement rather
than delay.
39
Bargaining With Outside Options
4.4
Consider two agents bargaining over the division of a
and Agent
^
and the value of 9
ai can be thought to be more
=
1
Agent 2 accepts the
receives the remaining
1
—
the
is
costly, so that
if
game
Agent
r.
1 offers
a share Wr to Agent
ends. Agent 2 receives the proposal, Wr, and Agent
Wt- If Agent 2 rejects the
offer,
=
negotiations continue until date r
Finally, as in Yildiz (2003), the agents are
y
she decides whether to take her
=
E^
Agent
1
>
[61]
assume that
incurs a cost c
assumed to be "optimistic,"
- E^
[0]
1,
We
Agent 2 believing that her outside option
is
0.
0.
higher than Agent
— and y parameterizes the extent of optimism
>
in the sense that
In other words, they differ in their expectations of 6 on the outside option of Agent 2
option
where
1,
under 6l-
likely
offer,
{0, 1},
dates, the agents
outside option, terminating the game, or wait for the next stage of the game.
delay
re
£ {a^, an}, where the signal
{stjt^i with sj
Bargaining follows a simple protocol: at each date
If
Between the two
unknown.
initially
is
observe an infinite sequence of public signals s
2.
There are two dates,
an outside option 6 6 {^Li^h} that expires at the end of date
2 has
< 9h <
Oi
dollar.
in this
1 's
—
-with
assessment of this outside
game.
We assume that the game form and beliefs are common knowledge and look for the subgameperfect equilibrium of this simple bargaining game.
By backward
1 is
E^
[9\p (s)]
<
induction, at date r
1,
for
1,
and hence she accepts an
=
therefore offers wi
=
E^
[9\p (s)]. If there is
any p{s), the value of outside option
offer
wi
and only
if
no agreement
if
wi
>
for
E'^ [9\p (s)].
Agent
Agent
2
at date 0, the continuation values of
the two agents are:
yi
which uses the
=
1
_
c
- E^
fact that there
is
no cost of delay
the agents will delay the agreement to date r
e2
Here, E^ [E^
[0|/9(s)]]
is
Agent
after observing the signals
s.
V^
and
[E^ {9\p (s)]]
If
I's
[9]
-
for
=
E^ [E^
1 if
=
Agent
1
2.
and only
[e\p (s)]]
expectation about
Agent
E^ [E^
>
[9\p (s)]]
=
E^
Since they have
[9]
1 dollar in total,
if
c.
how Agent
2 will
update her
beliefs
expects that the information will reduce Agent 2's
40
expectation of her outside option more than the cost of waiting, then Agent
This description makes
on Agent
I's
When
it
clear that
assessment of
each agent
is
whether there
how Agent
will
wilhng to wait.
1 is
be agreement at date r
=
depends
2 will interpret the (public) signals.
certain about the informativeness of the signals, they agree ex ante
Lemma 4 in
that they will interpret the information correctly. Consequently, as in
the previous
subsection. Agent I's Bayesian updating wall indicate that the public information will reveal
him
Yildiz (2004) has
to be right.
shown that
"wait to persuade" Agent 2 that her outside option
that each agent
may
i.e.,
differ.
i is
certain that Pr*
Then, from Theorem
Pr' (E2 {9\p{s)]
^
9)
=
1
1,
for
delays the agreement to date r
=
—
(sj
9\9)
Agent
this reasoning gives
is
— p^ >
relatively low.
More
1/2 for some p^ and
an incentive to
1
specifically,
p-^,
assume
where p^ and p^
the agents agree that Agent 2 will learn her outside option,
each
1 if
i.
Hence, E^ [E^ [9\p{s)]]
and only
=
E^
[6].
Therefore, Agent
1
if
y>c,
i.e.,
and only
if
the level of optimism
if
is
higher than the cost of waiting.
This discussion
therefore indicates that the arrival of public information can create a reason for delay in
bargaining games.
We now
motive
show that when agents
for delay
is
are uncertain about the informativeness of the signals, this
reduced and there can be immediate agreement.
understands that the same signals
will
Intuitively, each agent
be interpreted differently by the other agent and thus
expects that they are less likely to persuade the other agent. This decreases the incentives to
delay agreement.
This result
is
illustrated starkly here, with
an example where a small amount of uncer-
tainty about the informativeness of signals removes
that the agents' beliefs are again as in
bility
[l
more than
—p—
(5/2, 1
1
— p^
—
-I-
e
Example
1
all
with
e
small.
Now Agent
to the event that that p (s) will be either in [p
(5/2]
,
inducing Agent 2 to stick to her prior.
that Agent 2 will not update her prior by much. In particular,
E^ [e2[0|p(s)]] =E^[9]
+
0{e).
[9]
- E^
[e2 [^Ip
(s)]]
41
= -O (e) <
c.
—
1
assigns proba-
6/2,p^
+
Hence, Agent
we have
Thus
E^
Suppose
incentives to delay agreement.
5/2] or in
1
expects
This implies that agents
will
agree at the beginning of the game. Therefore, the same forces that
led to delayed asset trading in the previous subsection
in bargaining
when agents
can also induce immediate agreement
are "optimistic"
Manipulation and Uncertainty
4.5
Our
final
example
how
intended to show
is
the pattern of uncertainty used in the
body
of the
paper can result from game theoretic interactions between an agent and an informed party,
example as
in cheap talk
this possibility,
games (Crawford and
we choose the simplest environment
discussion to the single agent setting
is
Sobel, 1982). Since our purpose
to
is
for
to illustrate
communicate these ideas and
limit the
—the generalization to the case with two or more agents
straightforward.
The environment
with a prior belief
decision x e
[0, 1],
tt
is
as follows.
G
(0, 1)
and
The
=
that
his payoff
—
is
state of the world
at
1
(x
—
=
t
0)
0.
G
is
At time
i
=
{0, 1},
1,
this agent has to
Thus the agent would
.
and the agent
like to
starts
make a
form as accurate
an expectation about Q as possible.
The other
=
s'
player
{sj}^j with
with
G
St
€
s\
The
{0, 1}.
a media outlet,
is
{0, 1},
M, which
observes a large (infinite) number of signals
and makes a sequence of reports to the agent
an exchangeable sequence, we can represent
are
as
denoted by
I's,
it
p'
G
[0, 1],
and
similarly s
it,
(tm: [0,1]^
A ([0,1])
is
puts probability
Otherwise,
i.e.,
if
[0, 1].
t^
i,
there
is
Since
s'
This
is
convenient
A ([0,1]),
[0,1].
Let
on the identity mapping, thus corresponding to
ctm
articles.
s'
media as a mapping
the set of probability distributions on
1
{s(}^j
as before, with the fraction of signals that
represented by p G
is
allows us to model the mixed strategy of the
where
=
reports s can be thought of as contents of newspaper articles, while
correspond to the information that the newspaper collects before writing the
is
s
i
be the strategy that
M
reporting truthfully.
manipulation (or misreporting) on the part of the media
outlet M.1'5
We
6
—
1
go (p)
'
also
assume
for simplicity that p' has a
and go when 9
=
>
p and go (p)
for all p
<
0,
such that
=
g\ {p)
=
for all p
continuous distribution with density gi
for ed\
>
p-
p
< p and
gi (p)
>
for all p
This assumption implies that
See Baron (2004) and Gentzkow and Shapiro (2006) for related models of media bias.
42
if
>
p,
when
while
M reports
truthfully,
om =
i.e.,
the superscript
of type
of
p'
,
a and
Let us
biased towards
is
and receives
now
(0, 1),
H — honest).
A/3
With
is
M (unobservable to the
honest and can only play a^j
probability Aq €
(0, 1
—
A//), the
=
=
and
— Xa — \h,
I
—
(where
is
outlet receives utility equal to x irrespective
x.
\
i
media outlet
manipulate the agent to choose high values of
equal to
utility
the media
Type a media
1.
like to
complementary probability
0,
6
A//
type
for
is
and hence would
towards
be asymptotic learning (and
suppose instead that there are three different types of player
With probabihty
agent).
will
agreement when there are more than one agent).
also asymptotic
Now
^ then Theorem 2 apphes and there
the media outlet
is
of type
j3
With
is
the
biased
x.
look for the perfect Bayesian equilibrium of the
game between the media
outlet
and the agent. The perfect Bayesian equilibrium can be represented by two reporting functions
a%;
:
[0, 1]
function
^A
(p
:
and
([0, 1])
[0,1]
-^
sequence of reports
cr^^
[Oil]i
is
:
[0, 1]
^A
([0, 1]) for
which determines the
belief of the agent that
and an action function x
p,
choice of the agent as a function of p (there
is
no
the two biased types of
:
[0,1]
—
>
[0,1],
loss of generality
M, and
updating
9=1
when
the
which determines the
here in restricting to pure
strategies).
In equilibrium, x
rule given o""^, a^,^
must be optimal
and the prior
tt;
for the agent given
and a"^ and
ct^^
(p;
<p
must be derived from Bayes
must be optimal
for the
two biased media
outlets given x.
Note
p',
first
without
that since the payoff to the biased media outlet does not depend on the true
loss of generality,
shght abuse of notation,
two types report
let
we can
restrict
<t^ (p) and
a^
a^
(p)
and a j^^ not
to
depend on
p'
.
Then, with a
be the respective densities with which these
p.
Second, the optimal choice of the agent after observing a sequence of signals with fraction
p being equal to
1 is
xip)^(l>
for all p
e
[0, 1], i.e.,
{p)
the agent will choose an action equal to his behef
(f>{p).
Third, an application of Bayes' rule implies the following belief for the agent:
if
4>{p)
=
p<p
(21)
{
TAHSl(p)+AaCr^(p)+A;30-M{P)
43
The
following
Lemma
p
lemma shows
that any (perfect Bayesian) equilibrium has a very simple form:
5 In any equilibrium, there exist
< p and
(p
=
(p)
Proof. Prom
(pA /'^^
(j)
exists Pi,p2
media type
(3
P
< P
=
But
{p).
(f)
It
such that
(p
(p)
=
(f)^
for
all
iov p
>
>
(f>{p)
<
't'iPi)-
p.
tt
when p >
Now
Since the media type
suppose that the lemma
Then we
in that case,
p.
also have
a^ (p^)
equation (21) implies that
constant over p e [0,p).
(l){p) is
—
--
(j)
The proof
is false
[pi)
since
—
0,
for (j){p)
analogous.
(p, 1] is
lemma
from
this
that p
^p,
follows immediately
and
=
(p)
contradicting the hypothesis. Therefore,
being constant over p e
p,
such that (piPi)
{p)
< n
(f>g
P-
we have cr^
{p),
minimizes x
>
n when p <
(21), 4>{p) <:
a maximizes x{p) =
and there
O'^l'
> n and
(j)^
that equilibrium beliefs will take the form given in
the next proposition:
Proposition 4 Suppose
then the unique equilibrium actions and beliefs are:
^M
x{p)
=
cp
(p)
(22)
a^(p)^ffo(p)
(23)
=
(24)
<^
^^fiSfe^
I
Proof. Consider the case p <
constant over p G
p) (by
[0,
this range. Since this range
^M ~
9o- Similarly,
a^^
=
p.
Lemma
is
the
As
5),
=
91 (P)
(P)
proof of
in the
^fp>-p-
Lemma
5,
equation (21) implies that
common
a^^ (p)
a^
support of the densities
gi. Substituting these equalities in (21),
a^
is
=
0.
Since
(j)
(p)
is
proportional to go on
and
50, it
we obtain
must be that
(24).
This proposition implies that the unique equilibrium of the game between the media outlet
and the agent leads to a
special case of our
model
of learning
under uncertainty. In particulaj,
the beliefs in (24) can be obtained by the appropriate choice of the functions /^
from equation
analyzed
is
(3) in
in this
Section
paper
is
2.
likely to
(•)
and fs
(•)
This illustrates that the type of learning under uncertainty
emerge in game-theoretic situations where one of the players
trying to manipulate the beliefs of others.
44
Concluding Remarks
5
A key assumption of most theoretical analyses is that individuals have a "common prior,"
game
ing that they have beliefs consistent with each other regarding the
•
and possible distributions of payoff-relevant parameters. This presumption
common
the argument that sufficient
mean-
forms, institutions,
is
often justified by
experiences and observations, either through individual
observations or transmission of information from others, will eliminate disagreements, taking
agents towards
common
known theorems
Dubins
This presumption receives support from a number of well-
priors.
in statistics
and economics,
for
example. Savage (1954) and Blackwell and
(1962).
Nevertheless, existing theorems apply to environments in which learning occurs under certainty, that
is,
individuals are certain about the
tions, individuals are not only learning
meaning of
different signals.
many
In
situa-
about a payoff-relevant parameter but also about the
interpretation of different signals. This takes us to the realm of environments where learning
takes place under uncertainty. For example,
many
signals favoring a particular interpretation
might make individuals suspicious that the signals come from a biased source.
learning in environments with uncertainty
full identification (in the
may
We
show that
lead to a situation in which there
standard sense of the term in econometrics and
situations, information will be useful to individuals but
may
not lead to
in
common
which two individuals with
different priors observe the
informative about some underlying parameter.
certainty, since individuals
same
Our environment
full learning.
probability distributions of both individuals have
also
consider an environment
infinite
is
will take
sequence of signals
one of learning under un-
have non-degenerate subjective probability distribution over the
likelihood of different signals given the values of the parameter.
tions),
We
priors (or asymptotic agreement).
lack of
In such
statistics).
This paper investigates the conditions under which learning under uncertainty
individuals towards
is
full
We
show that when subjective
support (or in fact under weaker assump-
they will never agree, even after observing the same infinite sequence of signals.
show that
this corresponds to a result of
will agree, before observing the
parameter
will
"agreement to eventually disagree"
common
understanding that more information
to similar beliefs for agents has important implications for a variety of
when
there
is
individuals
sequence of signals, that their posteriors about the underlying
not converge. This
models. Instead,
;
We
no
full
may
not lead
games and economic
support in subjective probably distributions, asymptotic
45
learning and agreement
An
may
obtain.
important implication of this analysis
is
that after observing the
same sequence
two Bayesian individuals may end up disagreeing more than they originally
signals,
result contrasts with the
common presumption
of
did. This
that shared information and experiences will
take individuals' assessments-^Ioser to each other.
We
also systematically investigate
whether asymptotic agreement obtain as the amount of
uncertainty in the environment diminishes
(i.e.,
as
we look
at families of subjective probability
distributions converging to degenerate limit distributions with
We
all
their
mass at one point).
provide a complete characterization of the conditions under which this will be the case.
Asymptotic disagreement may prevail even under "approximate
certainty," as long as the family
of subjective probability distributions converging to a degenerate distribution (and thus to an
environment with certainty) has regularly-varying
or the t-distributions)
tails
(such as for the Pareto, the log-normal
In contrast, with rapidly- varying tails (such as the normal and the
.
exponential distributions), convergence to certainty leads to asymptotic agreement.
Lack of common
ior in
and common priors has important implications
a range of circumstances.
We
economic behavior
interacts with
tion,
beliefs
games of common
illustrate
games
interest, bargaining, asset trading
show how the
possibility of observing the
lay" in asset trading
among
example
why
illustrating
may
of signals
may
individuals that start with similar beliefs.
individuals
may be
of coordina-
For
wish to play common-
more information about
same sequence
games
and games of communication.
results, individuals
before rather than after receiving
economic behav-
the type of learning outhned in this paper
in various different situations, including
example, we show that contrary to standard
interest
how
for
payoffs. Similarly,
we
lead to "speculative de-
We also provide a simple
uncertain about informativeness of signals
—the
strategic behavior of other agents trying to manipulate their beliefs.
The
issues raised here have important implications for statistics
as learning in game-theoretic situations.
corresponds to one in which there
are well-behaved
these posteriors
and converge
is
is
well
As noted above, the environment considered here
lack of full identification. Nevertheless, Bayesian posteriors
to a limiting distribution.
Studying the limiting properties of
more generally and how they may be used
econometric models
and econometrics as
an interesting area
for research.
46
for inference in under-identified
Appendix: Omitted Proofs
6
Proof of Theorem
Under the hypothesis
1.
of the
theorem and with the notation
in (2),
we have
l-2r„/n'
which converges to
or oo depending on Mmn^aofn/n is greater than 1/2 or less than 1/2.
If
lim„_oorn (s) /n > 1/2, then by (2), limn^co'Pl (s) = hm„_.c>o (pl (s) = 1, and if lim„_,oor„ (s) /n < 1/2,
then lim„_,oo'An (•s) = liin„_,oo "An (*) ~ 0. Since lim^_oo^n (s) /n = 1/2 occurs with probability zero,
this shows the second part. The first part follows from the fact that, according to each i, conditional
on 6 = A, lim„_,oor„ (s) /n = p^ > 1/2.
Lemma
Proof of
The proof
3.
Proof of Theorem
identical to that of
is
1.
6.
Lemma
(Parti) This part immediately follows from
as each TT\>f^k' {p{s))
3,
is
and
positive,
is finite.
^ifA^ipis))
Assume Fg = Fg
(Part 2)
only
Lemma
—
if (Tfc (tt^)
T^
(tt^))
Tfc
(
Then, by
each 9 e @.
for
=
(/e(p))gg@)
The
0.
Lemma
3,
(pl^ao (p)
~
"Afc.oolp)
has probabiUty
latter inequality
=
if
and
under both
probability measures Pr^ and Pr^ by hypothesis.
Proof of Theorem
s
(1,
kI,oo (P is))
is
.
1)
,
.
.
-
<t>ioc
continuous in p
Now,
of
7f
tt
=
(1/A',
.
.
,
.
=
.lf\Ap{s))
i^lf\.[p{s))
where 1
Define
7.
l/K).
'
V'
=
First, take n^
{^^' (P(^))).eej
tt-
=
- T.
Then,
7f.
[{fe (p(^))),,ej J
^
«,
and the inequality follows by the hypothesis of the theorem. Hence, by Lemma 3,
for each p {s) e [0, 1]. Since [0, 1] is compact and [0^,^ (p (s)) - 0^_^ (p {s))\
(P (s))| >
,
(s),
there exists
since I^J-oq (p(s))
-
e
>
such that
0fc^oo (p(s))|
\(i>l^oo
continuous in
is
(P (s))
~
tt^
and
for
each
0fc,oo
tt^,
(P (5))|
>
each p
^ for
(s)
E
there exists a neighborhood
[0, 1].
N {n)
such that
I^I.c^oIpIs)) -<Afc,oo(p(s))|
for all n^
,
tt'^
€
N
{ft).
Since Pr' (5)
Proof of Theorem
Lemma
8.
=
1,
Our proof
>
kfc-TT^I
A;
=
1,
..., /-C
and
s
G
5
the last statement in the theorem follows.
utilizes the following
two lemmas.
A.
lim
m^oo 0loo,m(P)
+ Zk'^k^RiP-Pih^'')^P-p(^,A>n)
^
Proof. By condition
(i),
liinm~>oo c
(i,
A'' ,m)
=
1
for
each
i
and
k.
Hence, for every distinct k and
k\
"^^°°
Then,
/V
vP)
Lemma A
"1-.00
c(i,A'-,m)
follows from
m^oo
Lemma
j
[m{p —
3.
47
p{i,A'')))
V
V
/
^
V
Lemma
p
with
(s)
>
B. For any i
(s)
\\p
-p
(i,
>
Q and h
(^))
-
I
Proof.
R
by hypothesis,
Since,
< K, and
k
each
II
|4,oo,m (P
Lemma
m> m,
rh such that for each
0, there exists
< h/m,
A'')
>
A, there exists h'
lim
m —*oo 0U,-
(P (^
^')
<
)
continuous at each {p{i,6)
is
^-
(25)
I
I
—
'p{j,9') ,p{i,9)
— p{j,9)), by
such that
0,
hm
- Hm
<f>l^^^ {p is))
<
A'))\
4>l,oo,m {P {h
e/2
(26)
I
and by condition
>
there exists rh
(iii),
h/h' such that
UU,-(P(^))- 1™ 4,^,„(p(s))|<£/2.
holds uniformly in
(Proof of Part
A
—p
(s)
||p
1) Since
A'^)
(i,
R
ip
<
||
h'
-p
A'')
(i,
=
implies that lim„^oo<?ifc,oo,m(p(^^''))
if
and only
latter holds
Part
if
if
limm^oo
only
m
oo
<Afc
(p
(*'
^'^))
R ip (i, A'^) - p
if
The
.
(i,
ij, A'''
A^'
=
j
,
j
and
each
for
k'
(27)
=/^
then imply
Since each ratio
'•
j
,p
-p
A'')
(i.
A'')]
(j.
is
7r;^//7r;^
=
(25).
k (by condition
Hence, lim^^oo (^I-.oo.m (p(^.^''))
1-
~
inequalities in (26)
(27)
"
by
each
^
k'
Lemma
"^i.oo.m (P (^' ^''')))
positive,
for
(i)),
Lemma
A, the
establishing
k,
L
(Proof of Part
Lemma
2) Fix
there exists
3,
holds for every
k'
^
k.
>
e
>
e'
Now, by
-p{i,
Pr^ (||p {s)
and
(i),
>
5
such that
0,
Fix also any
0.
there exists
II
>
/io,fc
< ho,k/m\e =
A'')
>
0fc_oo,m (p(*))
1
and
i
—
e
Since each
7rj.,/7ri-
is finite,
by
e'
such that
0,
= f
A'')
k.
whenever /'^, {p{s)) / f\k (p(s)) <
f {x)dx
>
-
{1
5)
Let
Qk,m
and K
=
t,
There
any
/
k'
f
min||3,||</(j,
e'K/2.
{p e
By
0.
(i),
A (L)
;
||p
-p
(i.
A'')
II
>
there exists hi^k
\\p{s)
—p
(i,A''
J
>
hi^k/Tn,
/(m(p(s)-p(i,A'=)))
Moreover, since limm^oo
,
m
)
such that, whenever
/c
(i,
^'^ to)
,
c
(i, ^,
<
m) =
1 for
each
2 for every k' -^ k
i
and
and
any
m
>
||x||
mi,fc,
>
/ii_fc,
/
(a;)
<
p{s) £ Qk,m, and
and
/(m(p(.)-p(z,^^-')))
c (i, a''
< ho,k/m}
exists a sufficiently large constant mi^k such that for
we have
k,
>
[x]
=
^,
m>
^
^1^^22
K
there exists
vfii^k
Tn2,fc
>
mi^fc
such that
This implies
/>(p(s))/A4p(^))<«',
establishing that
4,oo,^(p(s))>l-eNow,
Then, by
such that
(28)
assume that R{p(i,9) —p{j,6') ,p{i,d) —p(j,0)) =
for each distinct 9 and 9'.
~
Lemma A, limm-.oo 'Pk oo m{P (*' ^'^)) '' ^^'^ hence by Lemma B, there exists ma^fc > TO2,fc
for each
> m^^k, p{s) 6 Qk,m,
for j 7^
i,
m
4>ioo,miPis))>l-e.
48
(29)
when
Notice that
we obtain
(28)
and
(29) hold,
we have H^iJcm
the desired inequaht)' for each
Pr^ {\\4>l,m
is)
-
4>l,m
<
is)\\
E
=
e)
(')
m > fh:
^''
~ 0L,m
-
(li-^-,- (^)
(s)||
<
<m
^-
Then, setting
(s)\\
<
^1^
=
m = naax^t rm^k,
A'^) Pt' [9
= A'
k<K
k<K
k<K
= 1-6.
(Proof of Part
9'.
fc.
Then, since each
3)
7r^.,/7r^ is
- p (j, 6') ,p{i,e)-p {j, 9)) j^ for each distinct 9 and
Lemma A imphes that limm_oo (Pi cxmiP (^' ^'^)) ^ ^ ^°^ ^ach
R (p
Assume that
positive,
{i,
9)
Let
K
Then, by part
•^fc.oo
(pC^s))
p{s) 6
>
2, for
1
each
/Tt
—'OO
J
m
there exists m2,fc such that for every
> m2,k and p [s) G Qk.m, we have
B, there also exists m^^k > Tn2,k such that for every
> m^^k and
k,
m
— £ By Lemma
Qfc,„j,
€.oo,m (P
(«))
< ^l™, 4,oc,m
This imphes that ||<p^,m (p(s))
IkL.m
I.
(s)
-
<Plo,-m {s)\\
<
e
— ^^.m
at the
(P
(pt''))!!
(i,
^'))
>
^-
+£<
Setting
1
tti
end of the proof of Part 2 to
the desired inequality.
49
-
2e
=
maxf^ms^fc and changing
< 4,^
||(/>^_„ (s)
(p (s))
- 0^_„
-
6.
(s)||
>
e,
we obtain
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